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.

翻译：我们针对具有非零边界数据的非线性初边值问题（IBVPs），推导了新的边界条件及其实现程序，使得解保持有界性。该新型边界处理流程被应用于包含耗散项的反对称形式非线性IBVPs。完整流程包含两个主要部分：第一部分（见文献[1,2]）针对含一阶导数的问题，通过边界项构建了以曲面积分表示的能量和熵变化率；本文第二部分通过引入二阶导数项及新型非线性边界程序进行补充，实现了对非零数据边界条件的处理。这种新型非线性边界程序将线性问题中广为人知的特征边界处理推广至非线性情形。为引入该流程，我们分析了涵盖线性对流方程和Burgers方程的标量反对称IBVP模型。完成连续分析后，我们证明通过结合求和积分算子与弱边界条件可得到能量稳定的非线性离散近似。随后将标量分析推广至一般非线性方程组。最后，将该新型边界程序应用于计算流体动力学中的四个重要IBVPs：不可压缩Euler方程、Navier-Stokes方程、浅水方程及可压缩Euler方程。