momentumsolvers.c

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#include "momentumsolvers.h"
 
/* Adaptive pseudo-CFL cap (see cfl_cap in MomentumSolver_DualTime_Picard_JamesonRK).
 * SAFETY: fraction of a failed CFL the controller may return to.
 * RELAX:  per-accepted-trial growth of the cap, so a transient rejection is not permanent. */
#define MOM_CFL_CAP_SAFETY 0.60
#define MOM_CFL_CAP_RELAX  1.005
 
 
/*================================================================================*
 *               SHARED PHYSICAL-TIME (BDF) COEFFICIENT PLUMBING                  *
 *================================================================================*/
 
#undef __FUNCT__
#define __FUNCT__ "MomentumUsesBDF2"
/**
 * @brief Single source of truth for the BDF1/BDF2 selection. See header.
 *
 * The run loop increments `step` before invoking the momentum solve.  Therefore,
 * BDF1 applies at a fresh start until two physical states exist. A committed
 * checkpoint restores `Ucont_rm1`, so a restarted solve can retain BDF2 on its
 * first advanced step.
 */
PetscBool MomentumUsesBDF2(SimCtx *simCtx)
{
    const PetscInt ti      = simCtx->step;
    const PetscInt tistart = simCtx->StartStep;
    return (PetscBool)(COEF_TIME_ACCURACY > 1.1 && ti != 1 &&
                       (simCtx->restartHistoryAvailable || ti > tistart + 1));
}
 
#undef __FUNCT__
#define __FUNCT__ "MomentumBDFCoefficient"
/**
 * @brief Returns a0 in {1.0, 1.5} for the current physical step. See header.
 */
PetscReal MomentumBDFCoefficient(SimCtx *simCtx)
{
    return MomentumUsesBDF2(simCtx) ? COEF_TIME_ACCURACY : 1.0;
}
 
 
#undef __FUNCT__
#define __FUNCT__ "ComputeTotalResidual"
/**
 * @brief Shared implementation of `ComputeTotalResidual()`.
 */
PetscErrorCode ComputeTotalResidual(UserCtx *user)
{
    PetscErrorCode ierr;
    SimCtx *simCtx = user->simCtx;
    
    // Extract Time Parameters from Context.
    // BDF step selection now lives in MomentumUsesBDF2() (shared with the stability estimate).
    const PetscReal dt      = simCtx->dt;
 
    PetscFunctionBeginUser;
    PROFILE_FUNCTION_BEGIN;
    // 1. Calculate Spatial Terms (stored in user->Rhs)
    //    Rhs = -Div(Flux) + Viscous + Source
    ierr = ComputeRHS(user, user->Rhs); CHKERRQ(ierr);
 
    // 2. Add Physical Time Derivative Terms (BDF Discretization)
    //    The equation solved is: dU/dtau = RHS_Spatial + RHS_Temporal
    
    /* BDF order + leading coefficient from the shared helpers (single source of truth,
       shared with the momentum stability estimate). a0 = 1.5 (==COEF_TIME_ACCURACY) for
       BDF2, 1.0 for BDF1; using a0 for the current-state term keeps the residual
       numerically identical to the historical inlined coefficients. */
    const PetscBool use_bdf2 = MomentumUsesBDF2(simCtx);
    const PetscReal a0       = MomentumBDFCoefficient(simCtx);
    if (use_bdf2) {
        // --- BDF2 (Second Order Backward Difference) ---
        // (a0*U^{n} - 2.0*U^{n-1} + 0.5*U^{n-2}) / dt = RHS_Spatial(U^{n})
        ierr = VecAXPY(user->Rhs, -a0/dt,  user->Ucont);     CHKERRQ(ierr);
        ierr = VecAXPY(user->Rhs, +2.0/dt, user->Ucont_o);   CHKERRQ(ierr);
        ierr = VecAXPY(user->Rhs, -0.5/dt, user->Ucont_rm1); CHKERRQ(ierr);
    } else {
        // --- BDF1 (First Order / Euler Implicit) ---
        // (a0*U^{n} - U^{n-1}) / dt = RHS_Spatial(U^{n}), with a0 = 1.0
        ierr = VecAXPY(user->Rhs, -a0/dt,  user->Ucont);     CHKERRQ(ierr);
        ierr = VecAXPY(user->Rhs, +1.0/dt, user->Ucont_o);   CHKERRQ(ierr);
    }
 
    // 3. Enforce Boundary Conditions on the Residual
    ierr = EnforceRHSBoundaryConditions(user); CHKERRQ(ierr);
 
    PROFILE_FUNCTION_END;
    PetscFunctionReturn(0);
}
 
 
#undef __FUNCT__
#define __FUNCT__ "Momentum_Solver_Explicit_RungeKutta4"
/**
 * @brief Internal helper implementation: `MomentumSolver_Explicit_RungeKutta4()`.
 * @details Local to this translation unit.
 */
PetscErrorCode MomentumSolver_Explicit_RungeKutta4(UserCtx *user, IBMNodes *ibm, FSInfo *fsi)
{
    PetscErrorCode ierr;
    (void)fsi;
    // --- Context Acquisition ---
    // Get the master simulation context from the first block's UserCtx.
    // This is the bridge to access all former global variables.
    SimCtx *simCtx = user[0].simCtx;
    PetscReal dt = simCtx->dt;
    //PetscReal st = simCtx->st;
    PetscInt  istage;
    PetscReal alfa[] = {0.25, 1.0/3.0, 0.5, 1.0};
 
    PetscFunctionBeginUser;
 
    PROFILE_FUNCTION_BEGIN;
 
    LOG_ALLOW(GLOBAL, LOG_INFO, "Executing explicit momentum solver (Runge-Kutta) for %d block(s).\n",simCtx->block_number);
 
    // --- 1. Pre-Loop Initialization (Legacy Logic) ---
    // This block prepares boundary conditions and allocates the RHS vector for all blocks
    // before the main RK loop begins. This logic is preserved from the original code.
    LOG_ALLOW(GLOBAL, LOG_DEBUG, "Preparing all blocks for Runge-Kutta solve...\n");
    for (PetscInt bi = 0; bi < simCtx->block_number; bi++) {
 
        ierr = ApplyBoundaryConditions(&user[bi]); CHKERRQ(ierr);
 
        /*
        // Immersed boundary interpolation (if enabled)
        if (simCtx->immersed) {
            LOG_ALLOW(LOCAL, LOG_DEBUG, "  Performing pre-RK IBM interpolation for block %d.\n", bi);
            for (PetscInt ibi = 0; ibi < simCtx->NumberOfBodies; ibi++) {
                // The 'ibm' and 'fsi' pointers are passed directly from FlowSolver
                ierr = ibm_interpolation_advanced(&user[bi], &ibm[ibi], ibi, 1); CHKERRQ(ierr);
            }
        }
        */
        
        // Allocate the persistent RHS vector for this block's context
        ierr = VecDuplicate(user[bi].Ucont, &user[bi].Rhs); CHKERRQ(ierr);
    }
    LOG_ALLOW(GLOBAL, LOG_DEBUG, "Pre-loop initialization complete.\n");
 
    // --- 2. Main Runge-Kutta Loop ---
    // The legacy code had an outer `pseudot` loop that only ran once. We preserve it.
    for (PetscInt pseudot = 0; pseudot < 1; pseudot++) {
        // Loop over each block to perform the RK stages
        for (PetscInt bi = 0; bi < simCtx->block_number; bi++) {
            for (istage = 0; istage < 4; istage++) {
                LOG_ALLOW(LOCAL, LOG_DEBUG, "  Block %d, RK Stage %d (alpha=%.4f)...\n", bi, istage, alfa[istage]);
 
                // a. Calculate the Right-Hand Side (RHS) of the momentum equation.
                ierr = ComputeRHS(&user[bi], user[bi].Rhs); CHKERRQ(ierr);
 
                // b. Advance Ucont to the next intermediate stage using the RK coefficient.
                //    Ucont_new = Ucont_old + alpha * dt * RHS
                ierr = VecWAXPY(user[bi].Ucont, alfa[istage] * dt, user[bi].Rhs, user[bi].Ucont_o); CHKERRQ(ierr);
 
                // c. Synchronize periodic endpoints and local ghosts for the new intermediate Ucont.
                const FieldId staggered_fields[] = {FIELD_ID_UCONT};
                ierr = SynchronizePeriodicStaggeredFields(&user[bi], 1, staggered_fields); CHKERRQ(ierr);
 
                // d. Re-apply boundary conditions for the new intermediate velocity.
                //    This is crucial for the stability and accuracy of the multi-stage scheme.
                ierr = ApplyBoundaryConditions(&user[bi]); CHKERRQ(ierr);
                
            } // End of RK stages for one block
 
            /*
            // Final IBM Interpolation for the block (if enabled)
            if (simCtx->immersed) {
                LOG_ALLOW(LOCAL, LOG_DEBUG, "  Performing post-RK IBM interpolation for block %d.\n", bi);
                for (PetscInt ibi = 0; ibi < simCtx->NumberOfBodies; ibi++) {
                    ierr = ibm_interpolation_advanced(&user[bi], &ibm[ibi], ibi, 1); CHKERRQ(ierr);
                }
            }
            */
 
        } // End loop over blocks
 
        // --- 3. Inter-Block Communication (Legacy Logic) ---
        // This is called after all blocks have completed their RK stages.
        if (simCtx->block_number > 1) {
      //    LOG_ALLOW(GLOBAL, LOG_DEBUG, "Updating multi-block interfaces after RK stages.\n");
        //   ierr = Block_Interface_U(user); CHKERRQ(ierr);
        }
 
    } // End of pseudo-time loop
 
    // --- 4. Cleanup ---
    // Destroy the RHS vectors that were created at the start of this function.
    for (PetscInt bi = 0; bi < simCtx->block_number; bi++) {
        ierr = VecDestroy(&user[bi].Rhs); CHKERRQ(ierr);
    }
    
    LOG_ALLOW(GLOBAL, LOG_INFO, "Runge-Kutta solve completed for all blocks.\n");
 
    PROFILE_FUNCTION_END;
 
    PetscFunctionReturn(0);
}
 
#undef __FUNCT__
#define __FUNCT__ "ComputeGlobalSpectralRadiusEstimate"
/**
 * @brief Compute a conservative global pseudo-time spectral radius estimate.
 *
 * The spectral radius lambda_max (units: 1/s) governs the pseudo-time stability limit
 * for the explicit Jameson 4-stage RK smoother. The pseudo-time step is then:
 *
 *     dtau = pseudo_cfl / lambda_max
 *
 * making pseudo_cfl a true dimensionless Courant number independent of the physical
 * timestep dt. Note: 2.83 is only the 4-stage RK *imaginary-axis* scalar limit; it is NOT
 * a generally-stable bound for the actual nonlinear/non-normal operator (see A4 validation).
 *
 * Per owned cell (k,j,i) the convective spectral radius contribution is:
 *
 *     lambda_cell = (|U_xi_flux| + |U_eta_flux| + |U_zeta_flux|) x (1/V)
 *
 * where:
 *   - U_xi_flux   = ucont[k][j][i].x : volumetric flux through the xi-face   [m3/s]
 *   - U_eta_flux  = ucont[k][j][i].y : volumetric flux through the eta-face   [m3/s]
 *   - U_zeta_flux = ucont[k][j][i].z : volumetric flux through the zeta-face  [m3/s]
 *   - 1/V = lAj[k][j][i]             : inverse cell volume (cell Jacobian)    [1/m3]
 *
 * Product units: [m3/s * 1/m3] = [1/s] = spectral radius.
 *
 * Only the 'positive' face at index (k,j,i) is used per direction (no neighbor
 * access), so no ghost-cell scatter is needed. lUcont is valid after the pre-loop
 * SynchronizePeriodicStaggeredFields; lAj is set at grid initialisation and never
 * changes. No DMGlobalToLocal calls are added.
 *
 * A BDF2 lower bound COEF_TIME_ACCURACY/dt is applied after the MPI reduction so
 * that zero-flow startup (all ucont == 0) gives dtau = pseudo_cfl * dt / 1.5 rather
 * than infinity.
 *
 * @param[in]  user         Array of UserCtx (one per block).
 * @param[in]  block_number Number of blocks.
 * @param[in]  dt           Physical time step (for the BDF2 lower bound).
 * @param[out] lambda_max_out  Global spectral radius [1/s].
 */
static PetscErrorCode ComputeGlobalSpectralRadiusEstimate(
    UserCtx *user, PetscInt block_number, PetscReal dt, PetscReal *lambda_max_out)
{
    PetscErrorCode  ierr;
    PetscReal       local_max = 0.0;
 
    PetscFunctionBeginUser;
 
    for (PetscInt bi = 0; bi < block_number; bi++) {
        DMDALocalInfo info = user[bi].info; /* pre-stored local ownership info */
        Cmpnts      ***ucont;
        PetscReal   ***aj;
 
        /* Read-only access: lUcont (valid after SynchronizePeriodicStaggeredFields),
           lAj (grid metric, unchanged after initialisation). */
        ierr = DMDAVecGetArrayRead(user[bi].fda, user[bi].lUcont, &ucont); CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(user[bi].da,  user[bi].lAj,    &aj);    CHKERRQ(ierr);
 
        /* Loop over owned cells only — no ghost-cell indices accessed. */
        for (PetscInt k = info.zs; k < info.zs + info.zm; k++) {
            for (PetscInt j = info.ys; j < info.ys + info.ym; j++) {
                for (PetscInt i = info.xs; i < info.xs + info.xm; i++) {
                    /* Sum absolute face-flux magnitudes (one face per coordinate direction).
                     * Using only the 'positive' face (index i/j/k, not i-1/j-1/k-1) avoids
                     * neighbor access. For smooth velocity fields the error is < 2x and the
                     * adaptive CFL controller handles any residual conservatism. */
                    PetscReal flux_sum = PetscAbsReal(ucont[k][j][i].x)
                                      + PetscAbsReal(ucont[k][j][i].y)
                                      + PetscAbsReal(ucont[k][j][i].z);
 
                    /* aj = 1/J = 1/cell_volume; lambda = flux/volume [1/s] */
                    PetscReal lambda = flux_sum * aj[k][j][i];
                    local_max = PetscMax(local_max, lambda);
                }
            }
        }
 
        ierr = DMDAVecRestoreArrayRead(user[bi].fda, user[bi].lUcont, &ucont); CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(user[bi].da,  user[bi].lAj,    &aj);    CHKERRQ(ierr);
    }
 
    /* Reduce across all MPI ranks to get the global maximum. */
    ierr = MPI_Allreduce(&local_max, lambda_max_out, 1, MPIU_REAL, MPI_MAX, PETSC_COMM_WORLD);
    CHKERRQ(ierr);
 
    /* BDF2 lower bound: if the velocity field is zero (startup, stagnant IC) lambda_max
       would be 0, giving dtau = inf. Clamp to 1.5/dt so dtau = pseudo_cfl * dt/1.5 in
       that degenerate case (conservative but finite). */
    *lambda_max_out = PetscMax(*lambda_max_out, COEF_TIME_ACCURACY / dt);
 
    PetscFunctionReturn(0);
}
 
 
/* Absolute metric-row sum for one face: |N.N| + |A.N| + |B.N|, where N is the
   face's own normal-metric vector and A,B are the other two face-metric vectors.
   Mirrors the g11/g21/g31 rows assembled per face in Viscous(). */
static inline PetscReal MomFaceGabs(Cmpnts N, Cmpnts A, Cmpnts B)
{
    const PetscReal NN = N.x*N.x + N.y*N.y + N.z*N.z;
    const PetscReal AN = A.x*N.x + A.y*N.y + A.z*N.z;
    const PetscReal BN = B.x*N.x + B.y*N.y + B.z*N.z;
    return PetscAbsReal(NN) + PetscAbsReal(AN) + PetscAbsReal(BN);
}
 
/* nvert bands matching the physical operator:
 *   SKIP    : cell excluded from the estimate (solid/blanked), nvert > 0.1.
 *   VISC1S  : Viscous() one-sided cross-derivative trigger band, (0.5, 7).
 *   QUICKBLK: Convection() QUICK-modifier band, [0.1, 7] (mirrors NOT(<0.1 || >innerblank)). */
#define MOM_SKIP_SOLID(v)    ((v) > 0.1)
#define MOM_VISC_ONESIDED(v) ((v) > 0.5 && (v) < 7.0)
#define MOM_QUICK_BLOCKS(v)  ((v) >= 0.1 && (v) <= 7.0)
 
/* Read-only: does ANY of the six viscous faces entering cell (k,j,i) use an
 * nvert-selected one-sided cross-derivative? Conservative solid-band check over
 * the full 3x3x3 neighborhood that the per-face Viscous() predicates can reach.
 * Indices i +/- 1, j +/- 1, k +/- 1 are always valid on the interior loop range. */
static PetscBool MomCellUsesOneSidedViscousStencil(PetscReal ***nvert, PetscInt k, PetscInt j, PetscInt i)
{
    for (PetscInt dk = -1; dk <= 1; dk++)
        for (PetscInt dj = -1; dj <= 1; dj++)
            for (PetscInt di = -1; di <= 1; di++) {
                if (!dk && !dj && !di) continue;
                if (MOM_VISC_ONESIDED(nvert[k+dk][j+dj][i+di])) return PETSC_TRUE;
            }
    return PETSC_FALSE;
}
 
/* Read-only: is cell (k,j,i) IB-adjacent (any solid in its 3x3x3 neighborhood)?
 * Uses the SKIP band so it is consistent with the one-sided viscous trigger. */
static PetscBool MomCellHasSolidNeighbor(PetscReal ***nvert, PetscInt k, PetscInt j, PetscInt i)
{
    for (PetscInt dk = -1; dk <= 1; dk++)
        for (PetscInt dj = -1; dj <= 1; dj++)
            for (PetscInt di = -1; di <= 1; di++) {
                if (!dk && !dj && !di) continue;
                if (MOM_SKIP_SOLID(nvert[k+dk][j+dj][i+di])) return PETSC_TRUE;
            }
    return PETSC_FALSE;
}
 
/* Read-only: is the QUICK reconstruction in one computational direction modified for
 * cell (k,j,i)? The cell uses BOTH faces in that direction (index f and f-1), whose
 * QUICK branches are gated by a non-periodic domain edge or by nvert blockers within
 * the stencil reach (including the second neighbor f-2). The +/-2 reads are guarded by
 * the ghost range [g0, g1) so they are safe for both periodic and non-periodic DMs. */
static PetscBool MomQuickDirModified(PetscReal ***nvert, PetscInt a, PetscInt b, PetscInt c,
                                     PetscInt idx, PetscInt m, PetscBool np0, PetscBool np1,
                                     PetscInt g0, PetscInt g1, char dir)
{
    /* (a,b,c) are the fixed indices; idx is the moving index in direction `dir`.
       The two contributing faces (idx, idx-1) have QUICK stencils spanning idx-2..idx+2. */
    if (np0 && idx <= 1)   return PETSC_TRUE;     /* faces idx,idx-1 reach the negative edge */
    if (np1 && idx >= m-2) return PETSC_TRUE;     /* ... or the positive edge */
    for (PetscInt d = -2; d <= 2; d++) {
        if (d == 0) continue;
        const PetscInt p = idx + d;
        /* Required stencil information unavailable in the local ghost range: do NOT assume
           fluid -> conservatively classify the direction as modified (use 2.5). */
        if (p < g0 || p >= g1) return PETSC_TRUE;
        PetscReal v;
        if      (dir == 'i') v = nvert[a][b][p];
        else if (dir == 'j') v = nvert[a][p][c];
        else                 v = nvert[p][b][c];
        if (MOM_QUICK_BLOCKS(v)) return PETSC_TRUE;
    }
    return PETSC_FALSE;
}
 
/* Active-row mask (3 bits: xi=1, eta=2, zeta=4) mirroring the final RHS masking.
 * A solid cell disables all rows; a positive solid neighbour or a positive non-periodic
 * physical face disables the corresponding normal (staggered) row; TwoD disables the
 * homogeneous direction. Returns 0 only when the location carries no active unknown. */
PetscInt MomCellActiveRows(PetscReal ***nvert, PetscInt k, PetscInt j, PetscInt i,
                           PetscInt mx, PetscInt my, PetscInt mz,
                           PetscBool np_x1, PetscBool np_y1, PetscBool np_z1, PetscInt twoD)
{
    if (MOM_SKIP_SOLID(nvert[k][j][i])) return 0;     /* solid cell: all rows inactive */
    PetscInt bits = 0x7;
    if (MOM_SKIP_SOLID(nvert[k][j][i+1])) bits &= ~0x1;  /* positive xi neighbour solid   */
    if (MOM_SKIP_SOLID(nvert[k][j+1][i])) bits &= ~0x2;  /* positive eta neighbour solid  */
    if (MOM_SKIP_SOLID(nvert[k+1][j][i])) bits &= ~0x4;  /* positive zeta neighbour solid */
    if (np_x1 && i == mx-2) bits &= ~0x1;                /* positive non-periodic xi face  */
    if (np_y1 && j == my-2) bits &= ~0x2;
    if (np_z1 && k == mz-2) bits &= ~0x4;
    if      (twoD == 1) bits &= ~0x1;                    /* TwoD homogeneous direction: xi  */
    else if (twoD == 2) bits &= ~0x2;                    /*                             eta  */
    else if (twoD == 3) bits &= ~0x4;                    /*                             zeta */
    return bits;
}
 
#undef __FUNCT__
#define __FUNCT__ "ComputeMomentumStabilityEstimate"
/**
 * @brief Practical conservative momentum pseudo-time stability estimate (shadow). See header.
 *
 * Operator-scaled estimate lambda = max_cell (a0/dt + lambda_c + lambda_nu) over active,
 * non-solid interior cells, blocks and MPI ranks (lambda_c already carries the per-direction
 * QUICK scheme factors -- it is NOT multiplied by f_c again here). Read-only; no halo exchange,
 * 5 global scalar collectives. NOT a proven spectral radius. Drives nothing in shadow mode.
 */
PetscErrorCode ComputeMomentumStabilityEstimate(UserCtx *user, PetscInt block_number,
                                                PetscReal dt, MomStabCandidate candidate,
                                                MomStabilityReport *rep)
{
    PetscErrorCode ierr;
    PetscMPIInt rank;
    PetscFunctionBeginUser;
 
    /* ---- input validation (no silent fallback) ---- */
    PetscCheck(user != NULL, PETSC_COMM_WORLD, PETSC_ERR_ARG_NULL, "user is NULL");
    PetscCheck(rep  != NULL, PETSC_COMM_WORLD, PETSC_ERR_ARG_NULL, "rep is NULL");
    PetscCheck(block_number > 0, PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "block_number must be > 0");
    PetscCheck(candidate >= MOM_STAB_CAND_B && candidate <= MOM_STAB_CAND_D,
               PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "unsupported stability candidate enum");
    PetscCheck(PetscIsNormalReal(dt) && dt > 0.0, PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE,
               "dt must be finite and positive");
 
    SimCtx         *simCtx   = user[0].simCtx;
    PetscCheck(simCtx != NULL, PETSC_COMM_WORLD, PETSC_ERR_ARG_WRONGSTATE, "user[0].simCtx is NULL");
    const PetscReal a0       = MomentumBDFCoefficient(simCtx);
    const PetscBool centered = (PetscBool)(simCtx->les || simCtx->central);
    const PetscBool has_nut  = (PetscBool)(simCtx->les || simCtx->rans);
    const PetscBool inviscid = (PetscBool)simCtx->invicid;
    PetscCheck(inviscid || (PetscIsNormalReal(simCtx->ren) && simCtx->ren > 0.0),
               PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Reynolds number must be finite and positive when viscous");
    const PetscReal lambda_t = a0 / dt;
    const PetscReal nu_mol   = inviscid ? 0.0 : 1.0 / simCtx->ren;
    const PetscInt  twoD     = simCtx->TwoD;
 
    /* Shadow-mode completeness flag. The estimate does NOT cover the Clark nonlinear stress
       Jacobian, nor RANS eddy-viscosity sign behaviour (not verified sign-definite).
       Body forces in the supported configs read a per-timestep-frozen scalar
       (simCtx->bulkVelocityCorrection) inside ComputeRHS, so within a pseudo-solve they are a
       constant forcing with ZERO velocity Jacobian (consistent with the frozen-pressure
       treatment) -- they do not make the estimate incomplete. */
    rep->estimate_incomplete = (PetscBool)(simCtx->clark || simCtx->rans);
 
    ierr = MPI_Comm_rank(PETSC_COMM_WORLD, &rank); CHKERRQ(ierr);
 
    /* Per-candidate local maxima; selected candidate also tracks the controlling cell. */
    PetscReal locmaxB = 0.0, locmaxC = 0.0, locmaxD = 0.0, sel_max = 0.0;
    PetscReal sel_lc = 0.0, sel_lv = 0.0;
    PetscInt  sel_ci = -1, sel_cj = -1, sel_ck = -1, sel_blk = -1, sel_class = 0, sel_os = 0;
    PetscInt  local_active = 0;
 
    for (PetscInt bi = 0; bi < block_number; bi++) {
        UserCtx *u = &user[bi];
        DMDALocalInfo info = u->info;
        const PetscInt mx = info.mx, my = info.my, mz = info.mz;
        const PetscInt xs = info.xs, xe = xs + info.xm;
        const PetscInt ys = info.ys, ye = ys + info.ym;
        const PetscInt zs = info.zs, ze = zs + info.zm;
        /* Interior range == where ComputeRHS evaluates the residual; boundary faces
           (0 and m-1) are RHS-masked, so excluding them implements active-row masking
           for the normal boundary rows. The last physical face (m-2) stays IN with the
           full cell estimate (its tangential rows remain active). */
        const PetscInt lxs = (xs==0)?xs+1:xs, lxe = (xe==mx)?xe-1:xe;
        const PetscInt lys = (ys==0)?ys+1:ys, lye = (ye==my)?ye-1:ye;
        const PetscInt lzs = (zs==0)?zs+1:zs, lze = (ze==mz)?ze-1:ze;
 
        /* Non-periodic boundary flags (QUICK stencil is degraded near these). */
        const PetscBool npx0 = (PetscBool)(u->boundary_faces[BC_FACE_NEG_X].mathematical_type != PERIODIC);
        const PetscBool npx1 = (PetscBool)(u->boundary_faces[BC_FACE_POS_X].mathematical_type != PERIODIC);
        const PetscBool npy0 = (PetscBool)(u->boundary_faces[BC_FACE_NEG_Y].mathematical_type != PERIODIC);
        const PetscBool npy1 = (PetscBool)(u->boundary_faces[BC_FACE_POS_Y].mathematical_type != PERIODIC);
        const PetscBool npz0 = (PetscBool)(u->boundary_faces[BC_FACE_NEG_Z].mathematical_type != PERIODIC);
        const PetscBool npz1 = (PetscBool)(u->boundary_faces[BC_FACE_POS_Z].mathematical_type != PERIODIC);
        const PetscBool wxn = (PetscBool)(u->boundary_faces[BC_FACE_NEG_X].mathematical_type == WALL);
        const PetscBool wxp = (PetscBool)(u->boundary_faces[BC_FACE_POS_X].mathematical_type == WALL);
        const PetscBool wyn = (PetscBool)(u->boundary_faces[BC_FACE_NEG_Y].mathematical_type == WALL);
        const PetscBool wyp = (PetscBool)(u->boundary_faces[BC_FACE_POS_Y].mathematical_type == WALL);
        const PetscBool wzn = (PetscBool)(u->boundary_faces[BC_FACE_NEG_Z].mathematical_type == WALL);
        const PetscBool wzp = (PetscBool)(u->boundary_faces[BC_FACE_POS_Z].mathematical_type == WALL);
 
        Cmpnts ***ucont, ***ucat = NULL;
        Cmpnts ***icsi, ***ieta, ***izet, ***jcsi, ***jeta, ***jzet, ***kcsi, ***keta, ***kzet;
        Cmpnts ***csi = NULL, ***eta = NULL, ***zet = NULL;
        PetscReal ***aj, ***iaj, ***jaj, ***kaj, ***nvert, ***nut = NULL;
 
        ierr = DMDAVecGetArrayRead(u->fda, u->lUcont, &ucont); CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->da,  u->lAj,   &aj);     CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->da,  u->lIAj,  &iaj);    CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->da,  u->lJAj,  &jaj);    CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->da,  u->lKAj,  &kaj);    CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lICsi, &icsi);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lIEta, &ieta);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lIZet, &izet);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lJCsi, &jcsi);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lJEta, &jeta);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lJZet, &jzet);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lKCsi, &kcsi);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lKEta, &keta);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lKZet, &kzet);   CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->da,  u->lNvert, &nvert); CHKERRQ(ierr);
        if (has_nut) { ierr = DMDAVecGetArrayRead(u->da, u->lNu_t, &nut); CHKERRQ(ierr); }
        /* Cartesian velocity + cell metrics: always read so candidate D's gradient term
           (and hence rep->lambda_D) is meaningful regardless of the selected candidate.
           Precondition: lUcat fresh (Contra2Cart has run in the preceding residual eval). */
        ierr = DMDAVecGetArrayRead(u->fda, u->lUcat, &ucat); CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lCsi,  &csi);  CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lEta,  &eta);  CHKERRQ(ierr);
        ierr = DMDAVecGetArrayRead(u->fda, u->lZet,  &zet);  CHKERRQ(ierr);
 
        /* Error flag for the cell loop: on a bad metric/estimate we record the location and
           jump to block_cleanup so EVERY acquired array is restored before the error returns. */
        PetscErrorCode cell_err = 0;
        PetscInt       be_i = -1, be_j = -1, be_k = -1;
        const char    *be_what = NULL;
 
        for (PetscInt k = lzs; k < lze; k++) {
            for (PetscInt j = lys; j < lye; j++) {
                for (PetscInt i = lxs; i < lxe; i++) {
                    /* ---- active-row mask: skip only locations with no active unknown ---- */
                    const PetscInt rows = MomCellActiveRows(nvert, k, j, i, mx, my, mz,
                                                            npx1, npy1, npz1, twoD);
                    if (rows == 0) continue;
                    local_active++;
                    const PetscReal Ajc = aj[k][j][i];
                    if (!(PetscIsNormalReal(Ajc) && Ajc > 0.0)) {
                        cell_err = PETSC_ERR_FP; be_i = i; be_j = j; be_k = k;
                        be_what = "non-finite/non-positive cell inverse-Jacobian";
                        goto block_cleanup;
                    }
 
                    /* ---- directional QUICK modification (per direction; both faces; 2nd neighbour) ---- */
                    const PetscBool mod_x = MomQuickDirModified(nvert, k, j, i, i, mx, npx0, npx1,
                                                               info.gxs, info.gxs+info.gxm, 'i');
                    const PetscBool mod_y = MomQuickDirModified(nvert, k, j, i, j, my, npy0, npy1,
                                                               info.gys, info.gys+info.gym, 'j');
                    const PetscBool mod_z = MomQuickDirModified(nvert, k, j, i, k, mz, npz0, npz1,
                                                               info.gzs, info.gzs+info.gzm, 'k');
 
                    /* ---- classify cell: interior / physical-boundary / IB-adjacent ---- */
                    const PetscBool bnd = (PetscBool)((npx0 && i<=1) || (npx1 && i>=mx-2) ||
                                                      (npy0 && j<=1) || (npy1 && j>=my-2) ||
                                                      (npz0 && k<=1) || (npz1 && k>=mz-2));
                    const PetscBool ib  = MomCellHasSolidNeighbor(nvert, k, j, i);
                    const PetscInt cls = ib ? 2 : (bnd ? 1 : 0);
 
                    /* ---- convective: six contravariant face fluxes, DIRECTIONAL QUICK factors ---- */
                    const PetscReal Uxp = ucont[k][j][i].x,   Uxm = ucont[k][j][i-1].x;
                    const PetscReal Uyp = ucont[k][j][i].y,   Uym = ucont[k][j-1][i].y;
                    const PetscReal Uzp = ucont[k][j][i].z,   Uzm = ucont[k-1][j][i].z;
                    const PetscReal divU = PetscAbsReal((Uxp-Uxm)+(Uyp-Uym)+(Uzp-Uzm));
                    const PetscReal fx = centered ? 1.0 : (mod_x ? 2.5 : (4.0/3.0));
                    const PetscReal fy = centered ? 1.0 : (mod_y ? 2.5 : (4.0/3.0));
                    const PetscReal fz = centered ? 1.0 : (mod_z ? 2.5 : (4.0/3.0));
 
                    /* per-direction transport scale; C's divergence term stays unscaled by f_d. */
                    const PetscReal lcB = 0.5 * Ajc * (
                          fx * (PetscAbsReal(Uxp)+PetscAbsReal(Uxm))
                        + fy * (PetscAbsReal(Uyp)+PetscAbsReal(Uym))
                        + fz * (PetscAbsReal(Uzp)+PetscAbsReal(Uzm)) );
                    const PetscReal lcC = lcB + 0.5 * Ajc * divU;
                    /* lambda_grad_u = max_i sum_j |du_i/dx_j| from fresh Cartesian velocity
                       (the nonlinear zero-order Jacobian term delta_u . grad u).
                       Physical gradient: d/dx_dir = Ajc*(Csi.dir d/dxi + Eta.dir d/deta + Zet.dir d/dzeta). */
                    const Cmpnts duc = { 0.5*(ucat[k][j][i+1].x-ucat[k][j][i-1].x),
                                         0.5*(ucat[k][j][i+1].y-ucat[k][j][i-1].y),
                                         0.5*(ucat[k][j][i+1].z-ucat[k][j][i-1].z) };
                    const Cmpnts due = { 0.5*(ucat[k][j+1][i].x-ucat[k][j-1][i].x),
                                         0.5*(ucat[k][j+1][i].y-ucat[k][j-1][i].y),
                                         0.5*(ucat[k][j+1][i].z-ucat[k][j-1][i].z) };
                    const Cmpnts duz = { 0.5*(ucat[k+1][j][i].x-ucat[k-1][j][i].x),
                                         0.5*(ucat[k+1][j][i].y-ucat[k-1][j][i].y),
                                         0.5*(ucat[k+1][j][i].z-ucat[k-1][j][i].z) };
                    const Cmpnts C = csi[k][j][i], E = eta[k][j][i], Z = zet[k][j][i];
                    #define MOM_ROW(cmp) ( \
                        PetscAbsReal(Ajc*(C.x*duc.cmp + E.x*due.cmp + Z.x*duz.cmp)) + \
                        PetscAbsReal(Ajc*(C.y*duc.cmp + E.y*due.cmp + Z.y*duz.cmp)) + \
                        PetscAbsReal(Ajc*(C.z*duc.cmp + E.z*due.cmp + Z.z*duz.cmp)) )
                    const PetscReal lgrad = PetscMax(MOM_ROW(x), PetscMax(MOM_ROW(y), MOM_ROW(z)));
                    #undef MOM_ROW
                    const PetscReal lcD = lcC + lgrad;
 
                    /* ---- viscous: six faces, face Jacobians, full metric rows ---- */
                    PetscReal lv = 0.0;
                    if (!inviscid) {
                        /* xi+ (i), xi- (i-1) */
                        for (PetscInt s = 0; s < 2; s++) {
                            const PetscInt fi = (s==0) ? i : i-1;
                            if (!(PetscIsNormalReal(iaj[k][j][fi]) && iaj[k][j][fi] > 0.0)) {
                                cell_err = PETSC_ERR_FP; be_i=i; be_j=j; be_k=k;
                                be_what = "non-finite/non-positive xi-face inverse-Jacobian"; goto block_cleanup; }
                            PetscReal nuf = nu_mol;
                            if (has_nut) {
                                PetscReal nt = 0.5*(nut[k][j][fi] + nut[k][j][fi+1]);
                                if ((wxn && fi==0) || (wxp && fi==mx-2)) nt = 0.0;
                                nuf += nt;
                            }
                            lv += PetscAbsReal(nuf) * Ajc * iaj[k][j][fi]
                                * MomFaceGabs(icsi[k][j][fi], ieta[k][j][fi], izet[k][j][fi]);
                        }
                        /* eta+ (j), eta- (j-1) */
                        for (PetscInt s = 0; s < 2; s++) {
                            const PetscInt fj = (s==0) ? j : j-1;
                            if (!(PetscIsNormalReal(jaj[k][fj][i]) && jaj[k][fj][i] > 0.0)) {
                                cell_err = PETSC_ERR_FP; be_i=i; be_j=j; be_k=k;
                                be_what = "non-finite/non-positive eta-face inverse-Jacobian"; goto block_cleanup; }
                            PetscReal nuf = nu_mol;
                            if (has_nut) {
                                PetscReal nt = 0.5*(nut[k][fj][i] + nut[k][fj+1][i]);
                                if ((wyn && fj==0) || (wyp && fj==my-2)) nt = 0.0;
                                nuf += nt;
                            }
                            /* eta-face normal is the eta metric (jeta) -> pass as N. */
                            lv += PetscAbsReal(nuf) * Ajc * jaj[k][fj][i]
                                * MomFaceGabs(jeta[k][fj][i], jcsi[k][fj][i], jzet[k][fj][i]);
                        }
                        /* zeta+ (k), zeta- (k-1) */
                        for (PetscInt s = 0; s < 2; s++) {
                            const PetscInt fk = (s==0) ? k : k-1;
                            if (!(PetscIsNormalReal(kaj[fk][j][i]) && kaj[fk][j][i] > 0.0)) {
                                cell_err = PETSC_ERR_FP; be_i=i; be_j=j; be_k=k;
                                be_what = "non-finite/non-positive zeta-face inverse-Jacobian"; goto block_cleanup; }
                            PetscReal nuf = nu_mol;
                            if (has_nut) {
                                PetscReal nt = 0.5*(nut[fk][j][i] + nut[fk+1][j][i]);
                                if ((wzn && fk==0) || (wzp && fk==mz-2)) nt = 0.0;
                                nuf += nt;
                            }
                            /* zeta-face normal is the zeta metric (kzet) -> pass as N. */
                            lv += PetscAbsReal(nuf) * Ajc * kaj[fk][j][i]
                                * MomFaceGabs(kzet[fk][j][i], kcsi[fk][j][i], keta[fk][j][i]);
                        }
                        lv *= 4.0;   /* 2 (two-face row-sum) x 2 (longitudinal full-stress) */
                    }
 
                    /* one-sided viscous cross-derivative branch near a solid: conservative x2,
                       applied ONCE per cell, after a complete 3x3x3 solid-band check. */
                    PetscInt one_sided = 0;
                    if (!inviscid && MomCellUsesOneSidedViscousStencil(nvert, k, j, i)) {
                        one_sided = 1;
                        lv *= 2.0;
                    }
 
                    /* ---- candidate totals and running maxima ---- */
                    const PetscReal lB = lambda_t + lcB + lv;
                    const PetscReal lC = lambda_t + lcC + lv;
                    const PetscReal lD = lambda_t + lcD + lv;
                    if (PetscIsInfOrNanReal(lD)) {
                        cell_err = PETSC_ERR_FP; be_i=i; be_j=j; be_k=k;
                        be_what = "non-finite stability estimate"; goto block_cleanup;
                    }
                    locmaxB = PetscMax(locmaxB, lB);
                    locmaxC = PetscMax(locmaxC, lC);
                    locmaxD = PetscMax(locmaxD, lD);
 
                    const PetscReal lsel  = (candidate==MOM_STAB_CAND_B)?lB:(candidate==MOM_STAB_CAND_C)?lC:lD;
                    const PetscReal lcsel = (candidate==MOM_STAB_CAND_B)?lcB:(candidate==MOM_STAB_CAND_C)?lcC:lcD;
                    if (lsel > sel_max) {
                        sel_max = lsel; sel_lc = lcsel; sel_lv = lv;
                        sel_ci = i; sel_cj = j; sel_ck = k; sel_blk = bi;
                        sel_class = cls; sel_os = one_sided;
                    }
                }
            }
        }
 
        /* Single cleanup point: reached on normal completion AND on any cell-loop error.
           Every successful GetArrayRead above has exactly one matching restore here. */
        block_cleanup:
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lUcont, &ucont); CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->da,  u->lAj,   &aj);     CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->da,  u->lIAj,  &iaj);    CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->da,  u->lJAj,  &jaj);    CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->da,  u->lKAj,  &kaj);    CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lICsi, &icsi);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lIEta, &ieta);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lIZet, &izet);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lJCsi, &jcsi);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lJEta, &jeta);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lJZet, &jzet);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lKCsi, &kcsi);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lKEta, &keta);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lKZet, &kzet);   CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->da,  u->lNvert, &nvert); CHKERRQ(ierr);
        if (has_nut) { ierr = DMDAVecRestoreArrayRead(u->da, u->lNu_t, &nut); CHKERRQ(ierr); }
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lUcat, &ucat); CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lCsi,  &csi);  CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lEta,  &eta);  CHKERRQ(ierr);
        ierr = DMDAVecRestoreArrayRead(u->fda, u->lZet,  &zet);  CHKERRQ(ierr);
 
        /* Now that this block's arrays are restored, surface any cell-loop error. */
        PetscCheck(cell_err == 0, PETSC_COMM_SELF, cell_err, "%s at (%" PetscInt_FMT
                   ",%" PetscInt_FMT ",%" PetscInt_FMT ")", be_what ? be_what : "error", be_i, be_j, be_k);
    }
 
    /* ---- global reductions (no ghost/halo exchange; scalar collectives only) ----
       5 collectives total: one 3-element MAX (B/C/D), one SUM (active-cell count),
       one MIN (portable owner selection), and two broadcasts (the controlling-cell
       real breakdown and integer indices). */
    PetscReal loc3[3] = { locmaxB, locmaxC, locmaxD }, glo3[3];
    ierr = MPI_Allreduce(loc3, glo3, 3, MPIU_REAL, MPI_MAX, PETSC_COMM_WORLD); CHKERRQ(ierr);
 
    PetscInt global_active = 0;
    ierr = MPI_Allreduce(&local_active, &global_active, 1, MPIU_INT, MPI_SUM, PETSC_COMM_WORLD); CHKERRQ(ierr);
 
    /* The selected candidate's global max equals glo3[candidate] (no extra reduction).
       Portable owner pick: lowest rank whose local sel_max equals the global max. */
    const PetscReal sel_global = glo3[(int)candidate];
    PetscMPIInt nproc, claim, owner;
    ierr = MPI_Comm_size(PETSC_COMM_WORLD, &nproc); CHKERRQ(ierr);
    claim = (sel_max == sel_global) ? rank : nproc;
    ierr = MPI_Allreduce(&claim, &owner, 1, MPI_INT, MPI_MIN, PETSC_COMM_WORLD); CHKERRQ(ierr);
    if (owner == nproc) owner = 0;   /* no active cell anywhere: deterministic fallback owner */
 
    PetscReal dbuf[2] = { sel_lc, sel_lv };
    PetscInt  ibuf[6] = { sel_ci, sel_cj, sel_ck, sel_blk, sel_class, sel_os };
    ierr = MPI_Bcast(dbuf, 2, MPIU_REAL, owner, PETSC_COMM_WORLD); CHKERRQ(ierr);
    ierr = MPI_Bcast(ibuf, 6, MPIU_INT,  owner, PETSC_COMM_WORLD); CHKERRQ(ierr);
 
    rep->lambda_t = lambda_t;
    rep->lambda_c = dbuf[0];
    rep->lambda_v = dbuf[1];
    rep->lambda   = sel_global;
    rep->lambda_B = glo3[0]; rep->lambda_C = glo3[1]; rep->lambda_D = glo3[2];
    rep->ci = ibuf[0]; rep->cj = ibuf[1]; rep->ck = ibuf[2];
    rep->cblock = ibuf[3]; rep->cclass = ibuf[4]; rep->one_sided = ibuf[5];
    rep->active_cells = global_active;
 
    /* Only a genuinely empty active set (all cells masked) may fall back to the temporal
       term alone. A non-finite or non-positive estimate with active cells is a hard error. */
    if (global_active == 0) {
        rep->lambda = lambda_t;
    } else {
        PetscCheck(PetscIsNormalReal(rep->lambda) && rep->lambda > 0.0, PETSC_COMM_WORLD, PETSC_ERR_FP,
                   "momentum stability estimate is non-finite or non-positive with %" PetscInt_FMT
                   " active cells", global_active);
    }
 
    /* Dominant limiter at the controlling cell. */
    if (rep->lambda_t >= rep->lambda_c && rep->lambda_t >= rep->lambda_v) rep->limiter = MOM_STAB_LIMITER_TIME;
    else if (rep->lambda_c >= rep->lambda_v)                              rep->limiter = MOM_STAB_LIMITER_CONVECTION;
    else                                                                  rep->limiter = MOM_STAB_LIMITER_VISCOSITY;
 
    PetscFunctionReturn(0);
}
 
 
#undef __FUNCT__
#define __FUNCT__ "MomentumSolver_DualTime_Picard_JamesonRK"
/**
 * @brief Internal helper implementation: `MomentumSolver_DualTime_Picard_JamesonRK()`.
 * @details Local to this translation unit.
 */
PetscErrorCode MomentumSolver_DualTime_Picard_JamesonRK(UserCtx *user, IBMNodes *ibm, FSInfo *fsi)
{
    (void)ibm;
    (void)fsi;
    // --- CONTEXT ACQUISITION BLOCK ---
    SimCtx *simCtx = user[0].simCtx;
    const PetscInt block_number = simCtx->block_number;
    
    // Legacy Names (Physics/Time) - Kept for brevity in formulas
    const PetscInt ti = simCtx->step;
    const PetscReal dt = simCtx->dt;       // Physical Time Step
    /* cfl: initial pseudo-CFL Courant number (dimensionless, flow-independent since Phase 3).
     * Actual dtau = cfl / lambda_max, computed after the spectral radius estimate below. */
    const PetscReal cfl = simCtx->pseudo_cfl;
    const PetscReal alfa[] = {0.25, 1.0/3.0, 0.5, 1.0}; // Jameson RK smoothing coefficients
    Vec *pRhs; // Per-block backup of Rhs at last accepted pseudo-state.
 
    // State flags
    PetscBool force_restart = PETSC_FALSE;
 
    // Renamed Solver Parameters
    const PetscInt  max_pseudo_steps = simCtx->mom_max_pseudo_steps; // Max Dual-Time Iterations
    const PetscReal tol_abs_delta    = simCtx->mom_atol; // Stop if |dU| < tol
    const PetscReal tol_rtol_delta   = simCtx->mom_rtol; // Stop if |dU|/|dU0| < tol
    // --- END CONTEXT ACQUISITION BLOCK ---
 
    PetscErrorCode ierr;
    PetscMPIInt    rank;
    PetscInt       istage, pseudo_iter;
    PetscInt       accepted_iter = 0, rejected_iter = 0, recovery_streak = 0;
    PetscReal      ts, te, cput;
 
    // --- Global Convergence Metrics ---
    PetscReal global_norm_delta = 10.0;
    PetscReal global_rel_delta  = 1.0;
    PetscReal global_norm_resid = 1.0;
    PetscReal global_rel_resid  = 1.0;
    PetscReal smoothed_trial_ratio = 1.0; /* EMA-smoothed ratio; initialized neutral */
 
    // --- Spectral-radius-based pseudo-time step ---
    /* lambda_max: global maximum spectral radius [1/s], computed once per physical timestep.
     * pseudo_dtau[bi]: physical pseudo-time step dtau = pseudo_cfl / lambda_max [time].
     *   Unlike the old scheme (dtau = pseudo_cfl * dt), this is independent of dt and makes
     *   pseudo_cfl a true Courant number. (2.83 is only the scalar imaginary-axis RK limit,
     *   not a generally-stable value for the actual operator -- to be characterized in A4.)
     * dtau_min / dtau_max: per-timestep bounds derived from simCtx->min/max_pseudo_cfl. */
    PetscReal  lambda_max = 0.0;
    /* cfl_cap: dimensionless memory of where trials have failed. Without it the controller
     * re-discovers the stability limit every physical step -- measured on the laminar
     * channel it climbed 1.36 -> 2.00, rejected, recovered, and climbed straight back, with
     * 28.8% of all trials spent above the limit. Carried in CFL (not dtau) so it stays
     * meaningful as lambda_max evolves; see the rejection and growth paths below. */
    PetscReal  cfl_cap = simCtx->max_pseudo_cfl;
    /* resid_ref: reference scale for the residual [same units as R], recomputed each
     * physical step. See the assembly below for why the absolute residual tolerance is
     * measured against this rather than being a raw dimensional bound. */
    PetscReal  resid_ref = 0.0;
    PetscReal  dtau_min, dtau_max;
 
    // --- Local Metric Arrays (Per Block) ---
    PetscReal *delta_sol_norm_init, *delta_sol_norm_prev, *delta_sol_norm_curr, *delta_sol_rel_curr;
    PetscReal *resid_norm_init,     *resid_norm_prev,     *resid_norm_curr,     *resid_rel_curr;
    PetscReal *pseudo_dtau;          /* per-block pseudo-time step dtau [physical time, NOT dtau/dt] */
    PetscReal *trial_ratio_log;        /* per-block step-to-step residual ratio (for diagnostics) */
    PetscReal  last_accepted_resid;    /* last accepted |R|, for final summary */
 
    PetscFunctionBeginUser;
    PROFILE_FUNCTION_BEGIN;
    ierr = MPI_Comm_rank(PETSC_COMM_WORLD, &rank); CHKERRQ(ierr);
    LOG_ALLOW(GLOBAL, LOG_INFO, "Executing Dual-Time Momentum Solver for %d block(s).\n", block_number);
 
    // --- Allocate metric arrays ---
    ierr = PetscMalloc2(block_number, &delta_sol_norm_init, block_number, &delta_sol_norm_prev); CHKERRQ(ierr);
    ierr = PetscMalloc2(block_number, &delta_sol_norm_curr, block_number, &delta_sol_rel_curr); CHKERRQ(ierr);
    ierr = PetscMalloc2(block_number, &resid_norm_init,     block_number, &resid_norm_prev); CHKERRQ(ierr);
    ierr = PetscMalloc2(block_number, &resid_norm_curr,     block_number, &pseudo_dtau); CHKERRQ(ierr);
    ierr = PetscMalloc1(block_number, &resid_rel_curr); CHKERRQ(ierr);
    ierr = PetscMalloc1(block_number, &trial_ratio_log); CHKERRQ(ierr);
    ierr = PetscMalloc1(block_number, &pRhs); CHKERRQ(ierr);
    last_accepted_resid = 0.0;
 
    ierr = PetscTime(&ts); CHKERRQ(ierr);
 
    // --- 1. Pre-Loop Initialization ---
    
    if (block_number > 1) {
      // ierr = Block_Interface_U(user); CHKERRQ(ierr);
    }
 
    for (PetscInt bi = 0; bi < block_number; bi++) {
        ierr = ApplyBoundaryConditions(&user[bi]); CHKERRQ(ierr);
        
        // Immersed boundary interpolation (if enabled)
 
        // Allocate workspace
        ierr = VecDuplicate(user[bi].Ucont, &user[bi].Rhs); CHKERRQ(ierr);
        ierr = VecDuplicate(user[bi].Rhs, &pRhs[bi]); CHKERRQ(ierr);
        ierr = VecDuplicate(user[bi].Ucont, &user[bi].dUcont); CHKERRQ(ierr);
        ierr = VecDuplicate(user[bi].Ucont, &user[bi].pUcont); CHKERRQ(ierr);
        
        // Initialize Backup (pUcont) with current state
        ierr = VecCopy(user[bi].Ucont, user[bi].pUcont); CHKERRQ(ierr);
 
        // --- Calculate Initial Total Residual (Spatial + Temporal) ---
        ierr = ComputeTotalResidual(&user[bi]); CHKERRQ(ierr);
 
        // Backup Rhs with current state
        ierr = VecCopy(user[bi].Rhs, pRhs[bi]); CHKERRQ(ierr);
 
        // Compute Initial Norms
        ierr = VecNorm(user[bi].Rhs, NORM_INFINITY, &resid_norm_init[bi]); CHKERRQ(ierr);
 
        /* Reference scale for the absolute residual test (see its use below). VecNorm is
         * collective, so this is already global; take the max across blocks. */
        {
            PetscReal ucont_inf;
            ierr = VecNorm(user[bi].Ucont, NORM_INFINITY, &ucont_inf); CHKERRQ(ierr);
            resid_ref = PetscMax(resid_ref, MomentumBDFCoefficient(simCtx) * ucont_inf / dt);
        }
        
        // Initialize history for backtracking logic
        resid_norm_prev[bi]     = resid_norm_init[bi]; /* meaningful ratio on first iteration */
        delta_sol_norm_prev[bi] = 1000.0;
        /* pseudo_dtau[bi] is set after ComputeGlobalSpectralRadiusEstimate below. */
 
        LOG_ALLOW(GLOBAL,LOG_INFO," Block %d | Max RHS = %.6f | initial pseudo-CFL = %.4f .\n", bi, resid_norm_init[bi], cfl);
    }
 
    /* --- Spectral Radius Estimate (computed once per physical timestep) ---
     * lambda_max [1/s] is the global maximum of (face_flux_sum * inverse_cell_volume)
     * over all owned cells across all MPI ranks. See ComputeGlobalSpectralRadiusEstimate().
     *
     * dtau = pseudo_cfl / lambda_max   (pseudo_cfl is now a true Courant number)
     * dtau_min/max are derived from the user-specified min/max_pseudo_cfl bounds.
     *
     * The BDF2 lower bound inside the helper ensures lambda_max >= 1.5/dt, so dtau
     * remains finite even when the velocity field is zero (startup / stagnant ICs). */
    ierr = ComputeGlobalSpectralRadiusEstimate(user, block_number, dt, &lambda_max); CHKERRQ(ierr);
    dtau_min = simCtx->min_pseudo_cfl / lambda_max;
    dtau_max = simCtx->max_pseudo_cfl / lambda_max;
 
    /* Initialise per-block pseudo-time step from the warm-start CFL (carried from last timestep). */
    for (PetscInt bi = 0; bi < block_number; bi++) {
        pseudo_dtau[bi] = cfl / lambda_max;   /* dtau [physical time], NOT a fraction of dt */
    }
 
    /* --- SHADOW MODE: new operator-scaled stability estimate (diagnostic only) ---
     * Computes the Workstream-A estimate alongside the legacy one and logs a compact
     * comparison. The legacy estimate above continues to drive production dtau; the
     * shadow estimate changes nothing. Enable with -mom_stability_shadow. */
    {
        PetscBool shadow = PETSC_FALSE;
        ierr = PetscOptionsGetBool(NULL, NULL, "-mom_stability_shadow", &shadow, NULL); CHKERRQ(ierr);
        if (shadow) {
            MomStabilityReport rep;
            ierr = ComputeMomentumStabilityEstimate(user, block_number, dt, MOM_STAB_CAND_C, &rep); CHKERRQ(ierr);
            const char *lim = (rep.limiter==MOM_STAB_LIMITER_TIME) ? "time"
                            : (rep.limiter==MOM_STAB_LIMITER_CONVECTION) ? "convection" : "viscosity";
            /* Detailed shadow comparison at DEBUG only (gated by level + function allowlist);
               never printed unconditionally. Enable in tests via the logging controls. */
            LOG_ALLOW(GLOBAL, LOG_DEBUG,
                "Momentum scale [shadow]: legacy_dtau=%.4e new_dtau=%.4e ratio=%.4f limiter=%s | "
                "lambda_legacy=%.4e lambda_new=%.4e (B=%.4e C=%.4e D=%.4e) | "
                "lt=%.4e lc=%.4e lv=%.4e | cell=(%d,%d,%d) blk=%d class=%d onesided=%d\n",
                cfl / lambda_max, cfl / rep.lambda, rep.lambda / lambda_max, lim,
                lambda_max, rep.lambda, rep.lambda_B, rep.lambda_C, rep.lambda_D,
                rep.lambda_t, rep.lambda_c, rep.lambda_v,
                (int)rep.ci, (int)rep.cj, (int)rep.ck, (int)rep.cblock, (int)rep.cclass, (int)rep.one_sided);
        }
    }
 
    LOG_ALLOW(GLOBAL, LOG_INFO,
        "Dual-time solver: lambda_max=%.4e [1/s]  resid_ref=%.4e  dtau_init=%.4e  dtau range [%.4e, %.4e]  "
        "CFL range [%.4f, %.4f]  rejection_threshold=%.3f (EMA alpha=%.2f)  "
        "growth=%.3f  reduction=%.3f  max_accepted=%d.\n",
        lambda_max, resid_ref, cfl / lambda_max, dtau_min, dtau_max,
        simCtx->min_pseudo_cfl, simCtx->max_pseudo_cfl,
        simCtx->mom_dt_jameson_residual_norm_noise_allowance_factor,
        simCtx->mom_ratio_ema_alpha,
        simCtx->pseudo_cfl_growth_factor, simCtx->pseudo_cfl_reduction_factor,
        max_pseudo_steps);
 
    // --- 2. Main Pseudo-Time Iteration Loop ---
    pseudo_iter = 0;
    PetscBool residual_convergence_enabled =
        (PetscBool)(simCtx->mom_resid_atol > 0.0 || simCtx->mom_resid_rtol > 0.0);
    PetscBool converged = PETSC_FALSE;
    PetscBool last_trial_nonfinite = PETSC_FALSE;
    /* Rejected iterations no longer consume the accepted-iteration budget.
       A hard cap of 3x max_pseudo_steps guards against infinite rejection loops. */
    const PetscInt max_total_attempts = max_pseudo_steps * 3;
    while (!converged && accepted_iter < max_pseudo_steps && pseudo_iter < max_total_attempts)
    {
        pseudo_iter++;
        force_restart = PETSC_FALSE;
 
        // Every attempt starts from the last globally accepted state.
        for (PetscInt bi = 0; bi < block_number; bi++) {
            ierr = VecCopy(user[bi].pUcont, user[bi].Ucont); CHKERRQ(ierr);
            ierr = VecCopy(pRhs[bi], user[bi].Rhs); CHKERRQ(ierr);
            const FieldId staggered_fields[] = {FIELD_ID_UCONT};
            ierr = SynchronizePeriodicStaggeredFields(&user[bi], 1, staggered_fields); CHKERRQ(ierr);
            ierr = ApplyBoundaryConditions(&user[bi]); CHKERRQ(ierr);
        }
    
        for (PetscInt bi = 0; bi < block_number; bi++) {
            
            // === 4-Stage Jameson RK Smoothing Loop ===
            for (istage = 0; istage < 4; istage++) {
                
                LOG_ALLOW(GLOBAL,LOG_TRACE," Pseudo-Iter: %d | RK-Stage: %d\n", pseudo_iter, istage);
                
                /* RK Update: U_new = U_old + (dtau * alpha) * Residual
                 * pseudo_dtau[bi] IS dtau [physical time] — no further multiplication by dt.
                 * (In the old scheme dtau = pseudo_cfl * dt; here dtau = pseudo_cfl / lambda_max.) */
                ierr = VecWAXPY(user[bi].Ucont,
                                pseudo_dtau[bi] * alfa[istage],
                                user[bi].Rhs,
                                user[bi].pUcont); CHKERRQ(ierr);
 
                // Sync Ghosts & Re-apply BCs for intermediate stage
                const FieldId staggered_fields[] = {FIELD_ID_UCONT};
                ierr = SynchronizePeriodicStaggeredFields(&user[bi], 1, staggered_fields); CHKERRQ(ierr);
                
                ierr = ApplyBoundaryConditions(&user[bi]); CHKERRQ(ierr);
 
                // --- Re-calculate Total Residual for next stage ---
                ierr = ComputeTotalResidual(&user[bi]); CHKERRQ(ierr);
      
            } // End RK Stages
 
            // Immersed boundary interpolation (if enabled)
 
            // === Convergence Metrics Calculation ===
            
            // Calculate dU = U_current - U_backup
            ierr = VecWAXPY(user[bi].dUcont, -1.0, user[bi].pUcont, user[bi].Ucont); CHKERRQ(ierr);
 
            // Compute Infinity Norms
            ierr = VecNorm(user[bi].dUcont, NORM_INFINITY, &delta_sol_norm_curr[bi]); CHKERRQ(ierr);
            ierr = VecNorm(user[bi].Rhs, NORM_INFINITY, &resid_norm_curr[bi]); CHKERRQ(ierr);
        
            // Normalize relative metrics
            if (pseudo_iter == 1) {
                delta_sol_norm_init[bi] = delta_sol_norm_curr[bi];
                delta_sol_rel_curr[bi]  = 1.0;
                resid_rel_curr[bi]      = 1.0;
                // resid_norm_init[bi] set correctly before the loop — do not overwrite
            } else {
                if (delta_sol_norm_init[bi] > 1.0e-10) 
                    delta_sol_rel_curr[bi] = delta_sol_norm_curr[bi] / delta_sol_norm_init[bi];
                else 
                    delta_sol_rel_curr[bi] = 0.0;
 
                if(resid_norm_init[bi] > 1.0e-10) 
                    resid_rel_curr[bi] = resid_norm_curr[bi] / resid_norm_init[bi];
                else 
                    resid_rel_curr[bi] = 0.0;
            }
          
            /* Pre-compute per-block step-to-step trial ratio for later file logging. */
            {
                const PetscReal resid_floor_log = 1.0e-30;
                if (resid_norm_prev[bi] > resid_floor_log)
                    trial_ratio_log[bi] = resid_norm_curr[bi] / resid_norm_prev[bi];
                else if (resid_norm_curr[bi] <= resid_floor_log)
                    trial_ratio_log[bi] = 0.0;
                else
                    trial_ratio_log[bi] = PETSC_MAX_REAL;
            }
        } // End loop over blocks
 
        // --- Update Global Convergence Criteria ---
        global_norm_delta = -1.0e20; 
        global_rel_delta  = -1.0e20; 
        global_norm_resid = -1.0e20;
        global_rel_resid  = -1.0e20;
        
        for (PetscInt bi = 0; bi < block_number; bi++) {
            global_norm_delta = PetscMax(delta_sol_norm_curr[bi], global_norm_delta);
            global_rel_delta  = PetscMax(delta_sol_rel_curr[bi],  global_rel_delta);
            global_norm_resid = PetscMax(resid_norm_curr[bi],      global_norm_resid);
            global_rel_resid  = PetscMax(resid_rel_curr[bi],      global_rel_resid);
        }
        ierr = PetscTime(&te); CHKERRQ(ierr);
        cput = te - ts;
        LOG_ALLOW(GLOBAL, LOG_INFO, "  Pseudo-Iter(k) %d: |dUk|=%e, |dUk|/|dU0| = %e, |Rk|/|R0| = %e, CPU=%.2fs\n",
              pseudo_iter, global_norm_delta, global_rel_delta, global_rel_resid, cput);
 
        // === Adaptive Pseudo-CFL Trial Acceptance and Rollback ===
        const PetscReal resid_floor = 1.0e-30;
        PetscReal global_trial_ratio = 0.0;
        PetscBool global_nonfinite = PETSC_FALSE;
        for (PetscInt bi = 0; bi < block_number; bi++) {
            PetscReal ratio;
            if (resid_norm_prev[bi] > resid_floor) {
                ratio = resid_norm_curr[bi] / resid_norm_prev[bi];
            } else if (resid_norm_curr[bi] <= resid_floor) {
                ratio = 0.0;
            } else {
                ratio = PETSC_MAX_REAL;
            }
            global_trial_ratio = PetscMax(global_trial_ratio, ratio);
            global_nonfinite = (PetscBool)(global_nonfinite ||
                PetscIsInfOrNanReal(delta_sol_norm_curr[bi]) ||
                PetscIsInfOrNanReal(resid_norm_curr[bi]) ||
                PetscIsInfOrNanReal(ratio));
        }
 
        PetscReal reduced_trial_ratio;
        PetscMPIInt local_nonfinite = global_nonfinite ? 1 : 0;
        PetscMPIInt reduced_nonfinite = 0;
        ierr = MPI_Allreduce(&global_trial_ratio, &reduced_trial_ratio, 1, MPIU_REAL, MPI_MAX, PETSC_COMM_WORLD); CHKERRQ(ierr);
        ierr = MPI_Allreduce(&local_nonfinite, &reduced_nonfinite, 1, MPI_INT, MPI_LOR, PETSC_COMM_WORLD); CHKERRQ(ierr);
        global_trial_ratio = reduced_trial_ratio;
        global_nonfinite = reduced_nonfinite ? PETSC_TRUE : PETSC_FALSE;
        last_trial_nonfinite = global_nonfinite;
 
        /* EMA-smooth the trial ratio to tolerate occasional non-monotonic residual bumps.
           Non-finite trials bypass smoothing and always trigger rollback.
 
           The smoothed value is only a CANDIDATE until the trial is accepted. A rejected
           trial is rolled back -- its state never happened -- so letting its ratio persist
           into the next decision is a state leak, and a costly one: measured on the laminar
           channel, a diverging trial at cfl 2.0 (raw ratio 3.97) pushed the EMA to 1.61, and
           the *next* trial at cfl 1.5 was rejected too despite a raw ratio of 0.994, i.e.
           despite the residual actually decreasing. That false rejection cost four discarded
           ComputeRHS calls and drove dtau down to cfl 1.125, below the measured optimum.
           The dtau reduction on the rejection path already carries the information; the EMA
           does not need to carry it as well. */
        PetscReal trial_smoothed = smoothed_trial_ratio;
        if (!global_nonfinite) {
            trial_smoothed = simCtx->mom_ratio_ema_alpha * global_trial_ratio
                           + (1.0 - simCtx->mom_ratio_ema_alpha) * smoothed_trial_ratio;
        }
        LOG_ALLOW(GLOBAL, LOG_DEBUG,
                  "    [k=%d] raw_ratio=%.4e smoothed_ratio=%.4e (threshold=%.3f) | "
                  "|R_prev|=%.6e | |R_curr|=%.6e | CFL=%.6f\n",
                  pseudo_iter, global_trial_ratio, trial_smoothed,
                  simCtx->mom_dt_jameson_residual_norm_noise_allowance_factor,
                  resid_norm_prev[0], resid_norm_curr[0], pseudo_dtau[0]);
 
        if (simCtx->no_pseudo_cfl_backtrack) {
            /* Diagnostic mode: commit every finite trial; only non-finite triggers rollback. */
            force_restart = global_nonfinite;
        } else {
            force_restart = (PetscBool)(global_nonfinite ||
                trial_smoothed > simCtx->mom_dt_jameson_residual_norm_noise_allowance_factor);
        }
 
        if (force_restart) {
            /* old_dtau: dtau used for this (rejected) trial [physical time].
             * next_dtau: reduced dtau for the retry, clamped to dtau_min.
             * Both are physical-time values; the effective CFL = dtau * lambda_max. */
            PetscReal old_dtau  = pseudo_dtau[0];
            PetscReal next_dtau = PetscMax(dtau_min, old_dtau * simCtx->pseudo_cfl_reduction_factor);
            /* This CFL failed: never grow back to it. MOM_CFL_CAP_SAFETY keeps the ceiling
             * off the edge, where the smoother damps worst even when it is still stable. */
            cfl_cap = PetscMax(simCtx->min_pseudo_cfl,
                               PetscMin(cfl_cap, old_dtau * lambda_max * MOM_CFL_CAP_SAFETY));
            rejected_iter++;
            recovery_streak = 0;
            for (PetscInt bi = 0; bi < block_number; bi++) {
                ierr = VecCopy(user[bi].pUcont, user[bi].Ucont); CHKERRQ(ierr);
                ierr = VecCopy(pRhs[bi], user[bi].Rhs); CHKERRQ(ierr);
                const FieldId staggered_fields[] = {FIELD_ID_UCONT};
                ierr = SynchronizePeriodicStaggeredFields(&user[bi], 1, staggered_fields); CHKERRQ(ierr);
                ierr = ApplyBoundaryConditions(&user[bi]); CHKERRQ(ierr);
                pseudo_dtau[bi] = next_dtau;
            }
            LOG_ALLOW(GLOBAL, LOG_WARNING,
                      "  Trial %d REJECTED (raw_ratio=%.4e, smoothed=%.4e, nonfinite=%d); "
                      "dtau %.4e -> %.4e (cfl_eff %.4f -> %.4f)%s\n",
                      pseudo_iter, global_trial_ratio, trial_smoothed, (int)global_nonfinite,
                      old_dtau, next_dtau, old_dtau * lambda_max, next_dtau * lambda_max,
                      (old_dtau == next_dtau) ? " [AT FLOOR — no dtau reduction]" : "");
            /* Log rejected trial to file before rolling back. */
            if (!rank) {
                for (PetscInt bi = 0; bi < block_number; bi++) {
                    FILE *f;
                    char filen[PETSC_MAX_PATH_LEN + 128];
                    ierr = PetscSNPrintf(filen, sizeof(filen),
                        "%s/Momentum_Solver_DualTime_Picard_Jameson_RK_History_Block_%1d.log",
                        simCtx->log_dir, bi); CHKERRQ(ierr);
                    if (simCtx->step == simCtx->StartStep + 1 && pseudo_iter == 1 && !simCtx->continueMode)
                        f = fopen(filen, "w");
                    else
                        f = fopen(filen, "a");
                    if (simCtx->continueMode && simCtx->step == simCtx->StartStep + 1 && pseudo_iter == 1)
                        PetscFPrintf(PETSC_COMM_WORLD, f, "# Continuation from step %" PetscInt_FMT "\n", simCtx->StartStep);
                    /* Log format: dtau [physical time] + cfl_eff [dimensionless Courant number = dtau*lambda_max] */
                    PetscFPrintf(PETSC_COMM_WORLD, f,
                        "Step: %d | PseudoIter(k): %d | dtau: %.6e | cfl_eff: %.4f | |dUk|: %le | "
                        "|dUk|/|dU0|: %le | |Rk|: %le | |Rk|/|R0|: %le | "
                        "trial_ratio: %le | smoothed_ratio: %le | status: rejected | "
                        "dtau_after: %.6e | cfl_eff_after: %.4f\n",
                        (int)ti, (int)pseudo_iter, old_dtau, old_dtau * lambda_max,
                        delta_sol_norm_curr[bi], delta_sol_rel_curr[bi],
                        resid_norm_curr[bi], resid_rel_curr[bi],
                        trial_ratio_log[bi], trial_smoothed,
                        next_dtau, next_dtau * lambda_max);
                    fclose(f);
                }
            }
            if (old_dtau <= dtau_min) {
                if (global_nonfinite) {
                    SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_CONV_FAILED,
                            "Momentum solver produced a non-finite trial at minimum pseudo-CFL.");
                }
                /* Finite rejection at the dtau floor: reducing further is impossible.
                   Break instead of retrying bit-identically until max_pseudo_steps. */
                LOG_ALLOW(GLOBAL, LOG_WARNING,
                          "  Trial %d REJECTED (ratio=%.4e) at minimum dtau (%.4e, cfl_eff=%.4f) "
                          "with no further reduction possible; breaking retry loop.\n",
                          pseudo_iter, global_trial_ratio, old_dtau, old_dtau * lambda_max);
                break;
            }
            continue;
        }
 
        accepted_iter++;
        smoothed_trial_ratio = trial_smoothed;   /* commit only for a trial that survives */
        last_trial_nonfinite = PETSC_FALSE;
        last_accepted_resid  = resid_norm_curr[0]; /* track for final summary */
        for (PetscInt bi = 0; bi < block_number; bi++) {
            ierr = VecCopy(user[bi].Ucont, user[bi].pUcont); CHKERRQ(ierr);
            ierr = VecCopy(user[bi].Rhs, pRhs[bi]); CHKERRQ(ierr);
            resid_norm_prev[bi] = resid_norm_curr[bi];
            delta_sol_norm_prev[bi] = delta_sol_norm_curr[bi];
        }
 
        /* old_dtau: dtau used for this (accepted) trial.
         * next_dtau: dtau for the next trial, grown/reduced based on convergence rate.
         * Bounds (dtau_min, dtau_max) are derived from simCtx->min/max_pseudo_cfl / lambda_max. */
        PetscReal old_dtau  = pseudo_dtau[0]; /* save before any update */
        PetscReal next_dtau = old_dtau;
        if (global_trial_ratio < 0.90) {
            /* Residual decreasing fast: grow dtau to accelerate convergence. */
            next_dtau *= simCtx->pseudo_cfl_growth_factor;
            recovery_streak = 0;
        } else if (global_trial_ratio <= 1.0) {
            /* Residual barely decreasing: grow only after 3 clean consecutive trials. */
            recovery_streak++;
            if (recovery_streak >= 3) {
                next_dtau *= simCtx->pseudo_cfl_growth_factor;
                recovery_streak = 0;
            }
        } else {
            /* Accepted but residual slightly grew (within EMA noise allowance): reduce dtau. */
            next_dtau *= simCtx->pseudo_cfl_reduction_factor;
            recovery_streak = 0;
        }
        /* Relax the cap slowly on healthy trials: a single rejection caused by a transient
         * must not hold the step down for the rest of the run, but recovery has to be slow
         * enough that the controller does not simply walk back into the wall. */
        cfl_cap = PetscMin(simCtx->max_pseudo_cfl, cfl_cap * MOM_CFL_CAP_RELAX);
        next_dtau = PetscMin(next_dtau, dtau_max);
        next_dtau = PetscMin(next_dtau, cfl_cap / lambda_max);
        next_dtau = PetscMax(next_dtau, dtau_min);
        for (PetscInt bi = 0; bi < block_number; bi++) pseudo_dtau[bi] = next_dtau;
 
        LOG_ALLOW(GLOBAL, LOG_DEBUG,
                  "  Trial %d ACCEPTED (raw_ratio=%.4e, smoothed=%.4e); |dU|=%.6e | "
                  "dtau %.4e -> %.4e (cfl_eff %.4f -> %.4f)\n",
                  pseudo_iter, global_trial_ratio, trial_smoothed, global_norm_delta,
                  old_dtau, next_dtau, old_dtau * lambda_max, next_dtau * lambda_max);
 
        /* --- Post-decision file logging (one row per accepted trial) --- */
        if (!rank) {
            for (PetscInt bi = 0; bi < block_number; bi++) {
                FILE *f;
                char filen[PETSC_MAX_PATH_LEN + 128];
                ierr = PetscSNPrintf(filen, sizeof(filen),
                    "%s/Momentum_Solver_DualTime_Picard_Jameson_RK_History_Block_%1d.log",
                    simCtx->log_dir, bi); CHKERRQ(ierr);
                if (simCtx->step == simCtx->StartStep + 1 && pseudo_iter == 1 && !simCtx->continueMode)
                    f = fopen(filen, "w");
                else
                    f = fopen(filen, "a");
                if (simCtx->continueMode && simCtx->step == simCtx->StartStep + 1 && pseudo_iter == 1)
                    PetscFPrintf(PETSC_COMM_WORLD, f, "# Continuation from step %" PetscInt_FMT "\n", simCtx->StartStep);
                /* Log format: dtau [physical time] + cfl_eff [dimensionless Courant number = dtau*lambda_max] */
                PetscFPrintf(PETSC_COMM_WORLD, f,
                    "Step: %d | PseudoIter(k): %d | dtau: %.6e | cfl_eff: %.4f | |dUk|: %le | "
                    "|dUk|/|dU0|: %le | |Rk|: %le | |Rk|/|R0|: %le | "
                    "trial_ratio: %le | smoothed_ratio: %le | status: accepted | "
                    "dtau_after: %.6e | cfl_eff_after: %.4f\n",
                    (int)ti, (int)pseudo_iter, old_dtau, old_dtau * lambda_max,
                    delta_sol_norm_curr[bi], delta_sol_rel_curr[bi],
                    resid_norm_curr[bi], resid_rel_curr[bi],
                    trial_ratio_log[bi], trial_smoothed,
                    next_dtau, next_dtau * lambda_max);
                fclose(f);
            }
        }
 
        /* --- Convergence decision ---
         *
         * The residual is what states that the momentum equations are satisfied at the
         * current state; the update norm measures progress, not correctness. The two
         * residual tests therefore carry different weight and are structured differently.
         *
         *   converged = residual_abs_pass || (residual_rel_pass && update_pass)
         *
         * ABSOLUTE branch, sufficient on its own. |R| <= mom_resid_atol * resid_ref says
         * the equations are satisfied to the configured level outright, so it needs no
         * corroboration -- this is what an absolute tolerance means, and how PETSc's SNES
         * and KSP already behave. Requiring the update guard alongside it made the floor
         * unable to fire: measured on the laminar channel at step 793, |R| dropped below
         * the floor at pseudo-iteration 8 and the step still ran to 20, because
         * |dU|/|dU0| had not yet reached its own tolerance.
         *
         * RELATIVE branch, weaker evidence, paired with the update guard. A 1000x
         * reduction of |R0| says the iteration made progress from wherever it started,
         * not that the result is small; |R0| itself collapses as a run approaches steady
         * state. The guard stays mandatory here.
         *
         * The update norm is never sufficient by itself in either branch. |dU| ~ dtau*|R|,
         * so it goes small whenever dtau goes small -- observed as |dU| falling only ~10%
         * over 100 pseudo-iterations of a completely frozen iteration. An ABSOLUTE bound
         * on it (mom_atol) is the disguised, step-size-dependent residual bound
         * |R| <= mom_atol/dtau, which tightens as the controller succeeds in growing dtau;
         * it duplicates the residual test while pulling against it and takes no part here.
         *
         * With no residual criterion configured at all -- both residual tolerances set
         * non-positive, an explicit opt-out since the defaults now enable them -- the
         * update norms are the only information available and the legacy test is retained
         * unchanged.
         */
        if (residual_convergence_enabled) {
            const PetscBool residual_abs_pass = (PetscBool)(
                simCtx->mom_resid_atol > 0.0 && resid_ref > 0.0 &&
                global_norm_resid <= simCtx->mom_resid_atol * resid_ref);
            const PetscBool residual_rel_pass = (PetscBool)(
                simCtx->mom_resid_rtol > 0.0 && global_rel_resid <= simCtx->mom_resid_rtol);
            const PetscBool update_pass = (PetscBool)(
                tol_rtol_delta <= 0.0 || global_rel_delta <= tol_rtol_delta);
            converged = (PetscBool)(residual_abs_pass || (residual_rel_pass && update_pass));
        } else {
            converged = (PetscBool)(global_norm_delta <= tol_abs_delta &&
                                    global_rel_delta <= tol_rtol_delta);
        }
 
        if (block_number > 1) {
             // ierr = Block_Interface_U(user); CHKERRQ(ierr);
        }
    } // End while loop
 
    if (last_trial_nonfinite) {
        SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_CONV_FAILED,
                "Momentum solver exhausted its attempt limit while recovering from a non-finite trial.");
    }
 
    /* Convert the final dtau back to a CFL number for the warm-start of the next physical timestep.
     * simCtx->pseudo_cfl stores a dimensionless Courant number (dtau * lambda_max); the next
     * timestep's lambda_max may differ as the flow evolves, so storing the CFL number (not dtau)
     * ensures the warm-start adapts correctly to the new spectral radius. */
    PetscReal next_dtau_start  = pseudo_dtau[0];
    PetscReal next_cfl_warmstart = next_dtau_start * lambda_max;  /* CFL = dtau * lambda_max */
 
    if (!rank) {
        LOG_ALLOW(GLOBAL, LOG_INFO,
                  "  Step %d finished: dtau=%.4e  cfl_eff=%.4f  lambda_max=%.4e [1/s]. "
                  "Next step warm-starts at cfl=%.4f.\n",
                  simCtx->step, next_dtau_start, next_cfl_warmstart, lambda_max, next_cfl_warmstart);
    }
 
    /* Store the CFL number (not dtau) so it remains meaningful across physical timesteps. */
    simCtx->pseudo_cfl        = next_cfl_warmstart;
    simCtx->mom_last_lambda_max = lambda_max;    
 
    /* Expose the last accepted finite state. On converged or floor-break exits pUcont==Ucont
       already (VecCopy would be an identity), but ghosts may have gone stale; always re-sync
       and re-apply BCs. Only restore the solution vector itself when nothing was ever accepted. */
    if (accepted_iter == 0) {
        for (PetscInt bi = 0; bi < block_number; bi++) {
            ierr = VecCopy(user[bi].pUcont, user[bi].Ucont); CHKERRQ(ierr);
            ierr = VecCopy(pRhs[bi], user[bi].Rhs); CHKERRQ(ierr);
        }
    }
    for (PetscInt bi = 0; bi < block_number; bi++) {
        const FieldId staggered_fields[] = {FIELD_ID_UCONT};
        ierr = SynchronizePeriodicStaggeredFields(&user[bi], 1, staggered_fields); CHKERRQ(ierr);
        ierr = ApplyBoundaryConditions(&user[bi]); CHKERRQ(ierr);
    }
 
    simCtx->mom_last_converged = converged;
    if (!converged) {
        PetscPrintf(PETSC_COMM_WORLD,
            "[WARNING] Momentum solver step %d: reached %d total attempts (%d accepted, %d rejected) "
            "without convergence; continuing from last accepted finite state.\n",
            (int)ti, pseudo_iter, accepted_iter, rejected_iter);
    }
    if (accepted_iter == 0) {
        PetscPrintf(PETSC_COMM_WORLD,
            "[WARNING] Momentum solver step %d: no pseudo-time trials were accepted; "
            "retaining physical-step entry state.\n", (int)ti);
    }
    LOG_ALLOW(GLOBAL, LOG_INFO,
              "Momentum solver finished: %d attempts (%d accepted, %d rejected) of %d max accepted / %d hard cap, "
              "converged=%s, last accepted |R|=%.6e, last accepted |dU|=%.6e, "
              "next_dtau=%.4e (cfl=%.4f).\n",
              pseudo_iter, accepted_iter, rejected_iter, max_pseudo_steps, max_total_attempts,
              converged ? "yes" : "no",
              (accepted_iter > 0) ? last_accepted_resid : resid_norm_init[0],
              (accepted_iter > 0) ? delta_sol_norm_prev[0] : 0.0,
              next_dtau_start, next_cfl_warmstart);
 
    // --- Final Cleanup ---
    for (PetscInt bi = 0; bi < block_number; bi++) {
        ierr = VecDestroy(&user[bi].Rhs); CHKERRQ(ierr);
        ierr = VecDestroy(&user[bi].dUcont); CHKERRQ(ierr);
        ierr = VecDestroy(&user[bi].pUcont); CHKERRQ(ierr);
        ierr = VecDestroy(&pRhs[bi]); CHKERRQ(ierr);
    }
    ierr = PetscFree(pRhs); CHKERRQ(ierr);
 
    ierr = PetscFree2(delta_sol_norm_init, delta_sol_norm_prev);CHKERRQ(ierr);
    ierr = PetscFree2(delta_sol_norm_curr, delta_sol_rel_curr);CHKERRQ(ierr);
    ierr = PetscFree2(resid_norm_init,     resid_norm_prev);CHKERRQ(ierr);
    ierr = PetscFree2(resid_norm_curr,     pseudo_dtau); CHKERRQ(ierr);
    ierr = PetscFree(resid_rel_curr); CHKERRQ(ierr);
    ierr = PetscFree(trial_ratio_log); CHKERRQ(ierr);
    
    PROFILE_FUNCTION_END;
    PetscFunctionReturn(0);
}