À¶Ý®ÊÓƵ

Jan Nordström, Professor Emeritus in Computational Mathematics at the Department of Mathematics, Linköping University

We first show that skew-symmetric formulations of the nonlinear terms in initial boundary value problems (IBVPs) leads to an energy rate in terms of surface integrals only. Next, we focus on the boundary conditions and present energy bounded strong and weak implementation procedures. The new boundary procedure generalizes the well-known characteristic boundary procedure for linear problems to the nonlinear setting. We exemplify the complete theory on a scalar skew-symmetric nonlinear IBVP including the linear advection and Burger’s equation. The scalar analysis is subsequently repeated for general nonlinear systems of equations and shown to hold for the shallow water, the incompressible and compressible Euler and Navier-Stokes equations and equations governing multi-phase flows in the volume of fluid setting. We conclude by indicating how the continuous analysis directly lead to nonlinear energy stability when using the SBP-SAT technique with summation-by-parts operators and weak boundary conditions.