A Unified Framework for Sponge-Layer Relaxation Methods and Damping Operators for Conservation Laws: Application to the Piston Problem of Gas Dynamics

TL;DR

This paper develops a unified framework for relaxation and damping methods to improve outflow boundary conditions in conservation laws, demonstrated on gas dynamics piston problem, reducing unphysical reflections.

Contribution

It introduces a novel relaxation method with matrix-valued weights and a unified framework linking various damping operators for conservation laws.

Findings

New relaxation method effectively absorbs outgoing waves.

Framework reveals relationships among different boundary treatment methods.

Application to gas dynamics shows improved boundary condition performance.

Abstract

This work addresses the imposition of outflow boundary conditions for one-dimensional conservation laws. While a highly accurate numerical solution can be obtained in the interior of the domain, boundary discretization can lead to unphysical reflections. We investigate and implement some classes of relaxation methods and far-field operators to deal with this problem without significantly increasing the size of the computational domain. We formulate these methods within a framework that allows to reveal relationships among them, and to propose novel extensions. In particular, we introduce a simple and robust relaxation method with a matrix-valued weight function that selectively absorbs outgoing waves. As a challenging model problem, we consider the Lagrangian formulation of the Euler equations for a polytropic gas with inflow boundary conditions determined by an oscillating piston.