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

Carlos Muñoz-Moncayo

TL;DR

This paper tackles outflow boundary reflections in 1D conservation laws by unifying relaxation-based and far-field damping absorbing boundary conditions within a sponge-layer framework. It introduces a matrix-weighted relaxation method (RM-M) that selectively damps outgoing waves and establishes theoretical links between RM, SDO, and Strang splitting, including nonlinear damping via the NDO approach. The methods are evaluated on the piston problem for the Lagrangian Euler equations of a polytropic gas, showing that directional damping in RM-M improves absorption in nonlinear regimes while linear cases preserve robustness and accuracy. The work provides a versatile boundary-treatment toolkit for 1D hyperbolic systems with practical implications for reducing domain size without compromising interior solution fidelity.

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.

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 tackles outflow boundary reflections in 1D conservation laws by unifying relaxation-based and far-field damping absorbing boundary conditions within a sponge-layer framework. It introduces a matrix-weighted relaxation method (RM-M) that selectively damps outgoing waves and establishes theoretical links between RM, SDO, and Strang splitting, including nonlinear damping via the NDO approach. The methods are evaluated on the piston problem for the Lagrangian Euler equations of a polytropic gas, showing that directional damping in RM-M improves absorption in nonlinear regimes while linear cases preserve robustness and accuracy. The work provides a versatile boundary-treatment toolkit for 1D hyperbolic systems with practical implications for reducing domain size without compromising interior solution fidelity.

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.
Paper Structure (15 sections, 57 equations, 7 figures, 3 tables, 3 algorithms)

This paper contains 15 sections, 57 equations, 7 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Sponge layer-based ABCs
  3. Slowing-down and damping far-field operators (SDO)
  4. The relaxation method (RM)
  5. Extensions of some sponge layer-based ABCs
  6. A relation between the RM and the far-field damping operator
  7. The RM with matrix-valued weight function (RM-M)
  8. On attaining higher order of accuracy in time with the RM
  9. Nonlinear damping far-field operator (NDO)
  10. The Euler equations
  11. The piston problem
  12. Numerical results
  13. Discretization details
  14. Parameter configuration
  15. Performance comparison

Figures (7)

  • Figure 1: Outgoing waves are absorbed in the sponge layer through the application of artificial boundary conditions.
  • Figure 2: A Gaussian pulse is initialized inside the sponge layer to represent the effect of nonlinear interactions taking place there. Thus, we consider its left-going component as a physical wave to be preserved. The right-going sawtooth-like wave corresponds to disturbances introduced by the motion of a piston, which will be described in Section \ref{['sec:piston problem']}. The sponge layer is colored light grey. Above: Effect of directional (SDO) and scalar (S-SDO) damping operators applied to the incoming physical wave. Below: Effect of directional relaxation method with matrix-valued weight function (RM-M) and scalar relaxation method (RM) applied to the incoming physical wave.
  • Figure 2: Relative error \ref{['eq: Error ABC']} for the nonlinear equations \ref{['eq:conservative form lagrangian']} with NDO ABC \ref{['sec:NDO']} and different damping functions $(\sigma=20,\,N=250)$
  • Figure 3: Weight functions $\Gamma_{(\cdot)}$ and damping functions $d_{(\cdot)}$, as defined in Equations \ref{['eq: Engsig-Karup weight function']}, \ref{['eq: Mayer weight function']}, and \ref{['eq: definition d and s SDO']} respectively
  • Figure 4: Relative error \ref{['eq: Error ABC']} for the nonlinear equations \ref{['eq:conservative form lagrangian']} with SDO ABC and different maximum damping rates $\sigma$$(N=250)$
  • ...and 2 more figures

Theorems & Definitions (1)

  • Remark 1