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.

翻译：我们针对具有非零边界数据的非线性初边值问题（IBVPs），推导出能确保解有界的新边界条件及实现流程。该新边界处理方法适用于包含耗散项的反对称形式非线性初边值问题。完整流程包含两大要素：第一部分（发表于文献[1,2]）中，我们针对含一阶导数的问题，建立了以边界项表征的曲面能量与熵率积分表达式；本第二部分则通过添加二阶导数耗散项并约束边界项加以完善。我们发展了非线性的特征边界处理方法，将线性问题的特征边界方法加以推广。本文同时考虑了强加与弱加两种非线性边界条件施加方式（针对非零边界数据），并证明了解的有界性。将该边界处理方法应用于计算流体动力学中四个重要的初边值问题：不可压缩欧拉方程与纳维-斯托克斯方程、浅水波方程及可压缩欧拉方程。最后，通过采用求和分部算子结合弱边界条件，验证了稳定离散逼近的可实现性。