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We derive new boundary conditions and implementation procedures for nonlinearinitial boundary value problems (IBVPs) with non-zero boundary data that leadto bounded solutions. The new boundary procedure is applied to nonlinear IBVPsin skew-symmetric form, including dissipative terms. The complete procedure hastwo main ingredients. In the first part (published in [1, 2]), the energy andentropy rate in terms of a surface integral with boundary terms was producedfor problems with first derivatives. In this second part we complement it byadding second derivative terms and new nonlinear boundary procedures leadingfor boundary conditions with non-zero data. The new nonlinear boundaryprocedure generalise the well known characteristic boundary procedure forlinear problems to the nonlinear setting. To introduce the procedure, askew-symmetric scalar IBVP encompassing the linear advection equation andBurgers equation is analysed. Once the continuous analysis is done, we showthat energy stable nonlinear discrete approximations follow by usingsummation-by-parts operators combined with weak boundary conditions. The scalaranalysis is subsequently repeated for general nonlinear systems of equations.Finally, the new boundary procedure is applied to four important IBVPs incomputational fluid dynamics: the incompressible Euler and Navier-Stokes, theshallow water and the compressible Euler equations.

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