On the conservation of finite difference WENO schemes in non-rectangular domains using the inverse Lax-Wendroff boundary treatments

Abstract

We discuss the issue of conservation of the total mass for finite difference WENO schemes solving hyperbolic conservation laws on a Cartesian mesh using the inverse Lax-Wendroff boundary treatments in arbitrary physical domains whose boundaries do not coincide with grid lines. The numerical fluxes near the boundary are suitably modified so that strict conservation of the total mass is achieved and the high order accuracy and non-oscillatory performance are not compromised. The key point is a suitable definition of the total mass, which is consistent with the high order accuracy finite difference framework over an arbitrary domain with a boundary not necessarily coinciding with grid lines. Extensive numerical examples are provided to demonstrate that our modified method is strictly conservative, and is high order accurate and has as good performance as the original high order WENO schemes with the Lax-Wendroff boundary treatments, for both smooth problems and problems with discontinuities, in both one- and two-dimensional problems involving both scalar equations and systems.