Nonlinear boundary conditions for initial boundary value problems with applications in computational fluid dynamics

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. Firstly, the energy rate in terms of a surface integral with boundary terms is derived. Secondly, we bound the surface integral by deriving new nonlinear boundary procedures for boundary conditions with non-zero data. The new nonlinear boundary procedure generalises 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.