The paper is concerned with the mathematical theory and numerical approximation of systems of partial differential equations (pde) of hyperbolic, pseudo-parabolic type. Some mathematical properties of the initial-boundary-value problem (ibvp) with Dirichlet boundary conditions are first studied. They include the weak formulation, well-posedness and existence of traveling wave solutions connecting two states, when the equations are considered as a variant of a conservation law. Then, the numerical approximation consists of a spectral approximation in space based on Legendre polynomials along with a temporal discretization with strong stability preserving (SSP) property. The convergence of the semidiscrete approximation is proved under suitable regularity conditions on the data. The choice of the temporal discretization is justified in order to guarantee the stability of the full discretization when dealing with nonsmooth initial conditions. A computational study explores the performance of the fully discrete scheme with regular and nonregular data.

翻译：本文关注双曲-伪抛物型偏微分方程系统的数学理论与数值逼近问题。首先研究了具有Dirichlet边界条件的初边值问题的若干数学性质，包括弱形式、适定性，以及当方程被视为守恒律变体时连接两个状态的行波解存在性。数值逼近部分采用基于Legendre多项式的空间谱逼近，结合具有强稳定性保持性质的时间离散格式。在数据满足适当正则性条件下证明了半离散逼近的收敛性。论证了时间离散格式的选择合理性，以确保处理非光滑初始条件时全离散格式的稳定性。通过计算研究探讨了全离散格式在正则与非正则数据下的性能表现。