Quinpi: Integrating stiff hyperbolic systems with implicit high order finite volume schemes

TL;DR

This work extends the Quinpi framework to stiff hyperbolic systems by combining a third-order space-time fully implicit discretization (CWENOZ reconstruction without ghost cells and a DIRK time integrator) with a predictor-corrector scheme that freezes nonlinear reconstruction weights, producing an effectively linear high-order implicit method for hyperbolic conservation laws. A flux-centered conservative time-limiting procedure based on numerical entropy production indicators (MOOD-inspired) detects troubled cells and replaces high-order fluxes with low-order fluxes to preserve conservation while suppressing oscillations. Numerical experiments on Euler gas dynamics, low-Mach, and scalar problems demonstrate that Quinpi achieves third-order accuracy on smooth regimes, robustly handles stiffness, and effectively limits time oscillations, enabling larger time steps focused on material wave propagation. The framework is problem-agnostic, mass-conserving, and supports future extensions to higher order and more efficient linear solvers, broadening the applicability of implicit high-order methods for stiff hyperbolic systems.

Abstract

Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g.~because the discrete system is nonlinear also on linear problems. This nonlinearity of the scheme is circumvented as proposed in (Puppo et al., Comm.~Appl.~Math.~\& Comput., 2023) for scalar conservation laws, where a first order implicit predictor is computed to freeze the nonlinear coefficients of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler's system of gas-dynamics.

Figures (9)

• Figure 1: The algorithm of the Quinpi scheme for the computation of the solution in a single time-step. The DIRK routine, which provides the high order solution with the space limiters computed by the predictor, is followed by the time limiter which downgrades the numerical fluxes at the interface of an irregular cell.
• Figure 2: Exact density solution (thin gray solid line) and wave speeds (thick lines) of the Riemann problems at final time.
• Figure 3: $L^1$-errors on the contact wave computed with the $\mathsf{Q}3_{{\mathcal{I}3}}$ schemes on several grids and different values of the threshold parameter $\gamma_2$.
• Figure 4: Comparison between third order implicit schemes, without time-limiting and with different time-limiting strategies. The left-most panel shows the density solution. The other panels show zooms on the three waves. The $\mathsf{Q}3$ solution of the colliding flows problem is not showed because it blows up before reaching the final time.
• Figure 5: Structure of the solutions in the $x$-$t$ diagram. Limited cell interfaces are marked at the time levels and locations where time-limiting is applied. The colors denote the loop number at which a cell interface is marked: interfaces limited at initial loops are marked in blue, while those limited at final loops are marked in red. These results can be compared with the data provided in Table \ref{['tab:stat']}.

Theorems & Definitions (6)

• Definition 1: Third order CWENOZ reconstruction, see CSV19:cwenoz
• Definition 2: Third order CWENOZ reconstruction without ghost cells, see STP23:cweno:boundary
• Definition 3: Third order CWENO-AO reconstruction, see SempliceVisconti:2020
• Remark 1
• Remark 2: Implicit reconstruction along characteristic variables.
• Definition 4