Multi-dimensional third-order time-implicit scheme for conservation laws

TL;DR

This paper introduces a multi-dimensional, third-order, time-implicit scheme for conservation laws that combines high-order spatial and temporal methods with limiting procedures to control oscillations, enabling larger time steps in stiff problems.

Contribution

It extends a previously proposed 1D implicit scheme to multi-dimensions, integrating a third-order WENO-Z reconstruction with a DIRK method and a novel time-limiting procedure based on entropy production.

Findings

Effective in reducing oscillations in implicit schemes.

Applicable to Euler equations in low Mach regimes.

Works on both structured and unstructured meshes.

Abstract

When dealing with stiff conservation laws, explicit time integration forces to employ very small time steps, due to the restrictive CFL stability condition. Implicit methods offer an alternative, yielding the possibility to choose the time step according to accuracy constraints. However, the construction of high-order implicit methods is difficult, mainly because of the non-linearity of the space and time limiting procedures required to control spurious oscillations. The Quinpi approach addresses this problem by introducing a first-order implicit predictor, which is employed in both space and time limiting. The scheme has been proposed in (Puppo et al., Comm. Comput. Phys., 2024) for systems of conservation laws in one dimension. In this work the multi-dimensional extension is presented. Similarly to the one-dimensional case, the scheme combines a third-order Central WENO-Z reconstruction in space with a third-order Diagonally Implicit Runge-Kutta (DIRK) method for time integration, and a low order predictor to ease the computation of the Runge-Kutta stages. Even applying space-limiting, spurious oscillations may still appear in implicit integration, especially for large time steps. For this reason, a time-limiting procedure inspired by the MOOD technique and based on numerical entropy production together with a cascade of schemes of decreasing order is applied. The scheme is tested on the Euler equations of gasdynamics also in low Mach regimes. The numerical tests are performed on both structured and unstructured meshes.