We present a strategy for interpreting nonlinear, characteristic-type penalty terms as numerical boundary flux functions that provide provable bounds for solutions to nonlinear hyperbolic initial boundary value problems with open boundaries. This approach is enabled by recent work that found how to express the entropy flux as a quadratic form defined by a symmetric boundary matrix. The matrix formulation provides additional information for how to systematically design characteristic-based penalty terms for the weak enforcement of boundary conditions. A special decomposition of the boundary matrix is required to define an appropriate set of characteristic-type variables. The new boundary fluxes are directly compatible with high-order accurate split form discontinuous Galerkin spectral element and similar methods and guarantee that the solution is entropy stable and bounded solely by external data. We derive inflow-outflow boundary fluxes specifically for the Burgers equation and the two-dimensional shallow water equations, which are also energy stable. Numerical experiments demonstrate that the new nonlinear fluxes do not fail in situations where standard boundary treatments based on linear analysis do.

翻译：我们提出一种策略，将非线性特征型惩罚项解释为数值边界通量函数，为具有开放边界的非线性双曲型初边值问题的解提供可证明的边界。该方法得益于近期研究发现如何将熵通量表达为由对称边界矩阵定义的二次型。该矩阵表述为系统化设计基于特征的边界条件弱施加惩罚项提供了额外信息。需要边界矩阵的特殊分解来定义一组合适的特征型变量。新边界通量可直接与高阶精确分裂形式间断伽辽金谱元法及类似方法兼容，并保证解具有熵稳定性且仅受外部数据约束。我们专门针对Burgers方程和二维浅水方程推导了流入-流出边界通量，这些通量同样具有能量稳定性。数值实验表明，在基于线性分析的标准边界处理方法失效的情况下，新的非线性通量仍能有效工作。