Nonlinear Boundary Conditions for Initial Boundary Value Problems with Applications in Computational Fluid Dynamics

TL;DR

This paper develops new nonlinear boundary conditions for initial boundary value problems, ensuring bounded solutions and energy stability, with applications to key equations in computational fluid dynamics.

Contribution

It introduces a generalized nonlinear boundary procedure extending characteristic methods to nonlinear IBVPs, including second derivatives and complex fluid dynamics equations.

Findings

Bounded solutions achieved for nonlinear IBVPs with new boundary conditions

Energy stable discrete approximations using summation-by-parts operators

Successful application to Euler, Navier-Stokes, shallow water, and compressible Euler equations

Abstract

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in skew-symmetric form, including dissipative terms. The complete procedure has two main ingredients. In the first part (published in [1, 2]), the energy and entropy rate in terms of a surface integral with boundary terms was produced for problems with first derivatives. In this second part we complement it by adding second derivative terms and new nonlinear boundary procedures leading for boundary conditions with non-zero data. The new nonlinear boundary procedure generalise the well known characteristic boundary procedure for linear problems to the nonlinear setting. To introduce the procedure, a skew-symmetric scalar IBVP encompassing the linear advection equation and Burgers equation is analysed. Once the continuous analysis is done, we show that energy stable nonlinear discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions. The scalar analysis is subsequently repeated for general nonlinear systems of equations. Finally, the new boundary procedure is applied to four important IBVPs in computational fluid dynamics: the incompressible Euler and Navier-Stokes, the shallow water and the compressible Euler equations.