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

TL;DR

Quinpi introduces an efficient framework for high order implicit schemes to solve stiff hyperbolic systems, combining predictor-corrector strategies and entropy-based limiting to improve stability and accuracy in challenging gas-dynamics problems.

Contribution

The paper develops a problem-independent, high order implicit scheme for stiff hyperbolic systems, utilizing a predictor to handle nonlinearity and a novel entropy-based time-limiting procedure.

Findings

Effective handling of stiff hyperbolic problems with high order schemes

Reduction of nonlinear solver complexity through predictor approach

Successful application to gas-dynamics simulations with improved stability

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.