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

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 on 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 dissipative terms and bound the boundary terms. We develop a new nonlinear boundary procedure which generalise the characteristic boundary procedure for linear problems. Both strong and weak imposition of the nonlinear boundary conditions with non-zero boundary data are considered, and we prove that the solution is bounded. The 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. Finally we show that stable discrete approximations follow by using summation-by-parts operators combined with weak boundary conditions.