Numerical boundary flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries

TL;DR

This paper introduces a systematic method for designing nonlinear boundary flux functions that ensure entropy stability and bounds for hyperbolic problems with open boundaries, compatible with high-order methods.

Contribution

It presents a novel matrix-based approach to construct entropy-stable boundary fluxes for nonlinear hyperbolic PDEs, including Burgers and shallow water equations.

Findings

New boundary fluxes guarantee entropy stability.

Fluxes are compatible with high-order spectral element methods.

Numerical tests show improved robustness over standard methods.

Abstract

We present a strategy for interpreting nonlinear, characteristic-type penalty terms as numerical boundary flux functions that provide provable bounds for solutions to nonlinear hyperbolic initial boundary value problems with open boundaries. This approach is enabled by recent work that found how to express the entropy flux as a quadratic form defined by a symmetric boundary matrix. The matrix formulation provides additional information for how to systematically design characteristic-based penalty terms for the weak enforcement of boundary conditions. A special decomposition of the boundary matrix is required to define an appropriate set of characteristic-type variables. The new boundary fluxes are directly compatible with high-order accurate split form discontinuous Galerkin spectral element and similar methods and guarantee that the solution is entropy stable and bounded solely by external data. We derive inflow-outflow boundary fluxes specifically for the Burgers equation and the two-dimensional shallow water equations, which are also energy stable. Numerical experiments demonstrate that the new nonlinear fluxes do not fail in situations where standard boundary treatments based on linear analysis do.