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：我们提出了一种针对多维半无界区域上双曲-抛物型问题的扩展间断伽辽金离散格式。基于前人在一维情形中的研究成果，我们将带状计算域划分为有界区域与无界子域：有界区域采用基于勒让德基函数的间断有限元进行离散，无界子域则以尺度化拉盖尔函数为基函数进行逼近。界面处的数值通量实现了两个区域的无缝耦合。数值试验表明，该耦合策略在处理线性和非线性标量及矢量模型问题时均能获得精确数值解。此外，在半无界区域部分可模拟高效吸收层，以抑制向外传播的信号，同时界面处反射杂波可忽略不计。通过调节拉盖尔基函数的尺度参数，该扩展DG格式能够在给定精度水平下，显著降低大空间尺度瞬态动力学模拟的计算成本，相较标准单域间断有限元技术具有明显优势。