We propose methods that augment existing numerical schemes for the simulation of hyperbolic balance laws with Dirichlet boundary conditions to allow for the simulation of a broad class of differential algebraic conditions. Our approach is similar to that of Thompson (1987), where the boundary values were simulated by combining characteristic equations with the time derivative of the algebraic conditions, but differs in two important regards. Firstly, when the boundary is a characteristic of one of the fields Thompson's method can fail to produce reasonable values. We propose a method of combining the characteristic equations with extrapolation which ensures convergence. Secondly, the application of algebraic conditions can suffer from $O(1)$ drift-off error, and we discuss projective time-stepping algorithms designed to converge for this type of system. Test problems for the shallow water equations are presented to demonstrate the result of simulating with and without the modifications discussed, illustrating their necessity for certain problems.

翻译：我们提出一些方法,用迪里切特边界条件来扩大模拟双曲平衡法的现有数字办法,以便模拟广泛的不同代数条件。我们的方法类似于汤普森(1987年)的方法,即边界值通过将特性方程式与代数条件的时间衍生物结合起来来模拟,但有两个重要方面不同。首先,当边界是汤普森方法之一的特征时,便无法产生合理的价值。我们建议一种方法,将特性方程式与外推法结合起来,以确保趋同。第二,代数条件的应用可能发生O(1)美元漂流错误,我们讨论旨在为这种系统聚合的预测性时间步算法。浅水方程式的试验问题是为了证明模拟和不经过讨论的修改的结果,说明它们对某些问题的必要性。