Approximate well-balanced WENO finite difference schemes using a global-flux quadrature method with multi-step ODE integrator weights

TL;DR

This paper introduces high-order well-balanced finite difference schemes for hyperbolic balance laws that integrate ODE solver weights directly into the flux computation, ensuring high accuracy for both steady and dynamic solutions.

Contribution

The novel approach embeds ODE integrator weights into the flux calculation, eliminating explicit ODE solving and achieving high-order steady-state accuracy in WENO schemes.

Findings

Achieves optimal convergence for time-dependent solutions.

Significantly reduces errors in steady-state solutions.

Compatible with up to 8th order ODE methods.

Abstract

In this work, high-order discrete well-balanced methods for one-dimensional hyperbolic systems of balance laws are proposed. We aim to construct a method whose discrete steady states correspond to solutions of arbitrary high-order ODE integrators. However, this property is embedded directly into the scheme, eliminating the need to apply the ODE integrator explicitly to solve the local Cauchy problem. To achieve this, we employ a WENO finite difference framework and apply WENO reconstruction to a global flux assembled nodewise as the sum of the physical flux and a source primitive. The novel idea is to compute the source primitive using high-order multi-step ODE methods applied on the finite difference grid. This approach provides a locally well-balanced splitting of the source integral, with weights derived from the ODE integrator. By construction, the discrete solutions of the proposed schemes align with those of the underlying ODE integrator. The proposed methods employ WENO flux reconstructions of varying orders, combined with multi-step ODE methods of up to order 8, achieving steady-state accuracy determined solely by the ODE method's consistency. Numerical experiments using scalar balance laws and shallow water equations confirm that the methods achieve optimal convergence for time-dependent solutions and significant error reduction for steady-state solutions.