Nonlinear Boundary Conditions for Energy and Entropy Stable Initial Boundary Value Problems in Computational Fluid Dynamics

TL;DR

This paper develops new nonlinear boundary conditions for hyperbolic PDEs in fluid dynamics, ensuring energy and entropy stability in numerical schemes through a systematic procedure applicable to key fluid equations.

Contribution

It introduces a general step-by-step method for deriving nonlinear boundary conditions that guarantee stability for hyperbolic problems in CFD.

Findings

The boundary conditions lead to energy and entropy bounded solutions.

The procedures are applicable to shallow water, incompressible, and compressible Euler equations.

Implementation on summation-by-parts schemes ensures stability.

Abstract

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems that lead to energy and entropy bounded solutions. A step-by-step procedure for general nonlinear hyperbolic problems on skew-symmetric form is presented. That procedure is subsequently applied to the three most important equations in computational fluid dynamics: the shallow water equations and the incompressible and compressible Euler equations. Both strong and weak imposition of the nonlinear boundary conditions are discussed. Based on the continuous analysis, we show that the new nonlinear boundary procedure lead to energy and entropy stable discrete approximations if the scheme is formulated on summation-by-parts form in combination with a weak implementation of the boundary conditions.