We introduce an extended discontinuous Galerkin discretization of hyperbolic-parabolic problems on multidimensional semi-infinite domains. Building on previous work on the one-dimensional case, we split the strip-shaped computational domain into a bounded region, discretized by means of discontinuous finite elements using Legendre basis functions, and an unbounded subdomain, where scaled Laguerre functions are used as a basis. Numerical fluxes at the interface allow for a seamless coupling of the two regions. The resulting coupling strategy is shown to produce accurate numerical solutions in tests on both linear and non-linear scalar and vectorial model problems. In addition, an efficient absorbing layer can be simulated in the semi-infinite part of the domain in order to damp outgoing signals with negligible spurious reflections at the interface. By tuning the scaling parameter of the Laguerre basis functions, the extended DG scheme simulates transient dynamics over large spatial scales with a substantial reduction in computational cost at a given accuracy level compared to standard single-domain discontinuous finite element techniques.

翻译：我们提出了一种用于多维半无界域上双曲-抛物问题的扩展间断伽辽金离散方法。基于一维情形的前期工作，我们将条带状计算区域分为一个有界区域（采用基于勒让德基函数的间断有限元离散）和一个无界子域（采用缩放拉盖尔函数作为基函数）。界面处的数值通量实现了两个区域的无缝耦合。在标量和矢量线性和非线性模型问题的数值测试中，该耦合策略能产生精确的数值解。此外，可在域的半无界部分模拟高效吸收层，以阻尼出射信号，同时界面处产生可忽略的虚假反射。通过调谐拉盖尔基函数的缩放参数，与标准单域间断有限元技术相比，扩展DG方案能在给定精度水平下以大幅降低的计算成本模拟大空间尺度上的瞬态动力学行为。