A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

TL;DR

This paper introduces a high-order explicit relaxation scheme combined with projective integration for efficiently solving nonlinear hyperbolic conservation laws across multiple dimensions.

Contribution

It develops a fully explicit, high-order relaxation method that handles stiffness and allows larger time steps using projective integration, applicable to various hyperbolic systems.

Findings

The scheme achieves CFL-like time step restrictions.

Inner steps are independent of the stiffness of the source term.

Numerical tests confirm stability and accuracy across multiple equations.

Abstract

We present a general, high-order, fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions. The scheme consists of two steps. In a first (relaxation) step, the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff BGK source term. Then, this kinetic equation is integrated in time using a projective integration method. After taking a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time step restriction on the outer time step is similar to the CFL condition for the hyperbolic conservation law. Moreover, the number of inner time steps is also independent of the stiffness of the BGK source term. We discuss stability and consistency, and illustrate with numerical results (linear advection, Burgers' equation and the shallow water and Euler equations) in one and two spatial dimensions.