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 nonlinear 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.

中文翻译：

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使用扩展 DG 方法在多维无界域上建立高效双曲-抛物线模型

我们引入了多维半无限域上双曲-抛物线问题的扩展不连续伽辽金离散化。基于之前在一维情况下的工作，我们将带状计算域分为有界区域和无界子域，其中使用勒让德基函数通过不连续有限元进行离散化，其中缩放拉盖尔函数用作基础。界面处的数值通量允许两个区域的无缝耦合。结果表明，所得到的耦合策略可以在线性和非线性标量和矢量模型问题的测试中产生准确的数值解。此外，可以在域的半无限部分中模拟有效的吸收层，以便抑制传出信号，而界面处的寄生反射可以忽略不计。通过调整拉盖尔基函数的缩放参数，扩展的 DG 方案可以模拟大空间尺度上的瞬态动力学，与标准单域不连续有限元技术相比，在给定精度水平下显着降低计算成本。