Stable semi-implicit SDC methods for conservation laws

TL;DR

This paper introduces semi-implicit spectral deferred correction methods inspired by Lax-Wendroff, achieving high-order, L-stability, and robustness for conservation laws with minimal diffusion, validated through extensive numerical experiments.

Contribution

Development of a family of semi-implicit SDC methods based on an implicit Lax-Wendroff formulation, offering high-order, L-stability, and computational efficiency for conservation laws.

Findings

Methods are L-stable up to order 11

Require minimal diffusion for orders 13 and 15

Numerical experiments confirm stability and accuracy across various conservation problems

Abstract

Semi-implicit spectral deferred correction (SDC) methods provide a systematic approach to construct time integration methods of arbitrarily high order for nonlinear evolution equations including conservation laws. They converge towards $A$- or even $L$-stable collocation methods, but are often not sufficiently robust themselves. In this paper, a family of SDC methods inspired by an implicit formulation of the Lax-Wendroff method is developed. Compared to fully implicit approaches, the methods have the advantage that they only require the solution of positive definite or semi-definite linear systems. Numerical evidence suggests that the proposed semi-implicit SDC methods with Radau points are $L$-stable up to order 11 and require very little diffusion for orders 13 and 15. The excellent stability and accuracy of these methods is confirmed by numerical experiments with 1D conservation problems, including the convection-diffusion, Burgers, Euler and Navier-Stokes equations.