波动方程的时空伽辽金边界元法
Space-Time Galerkin Boundary Element Method for the Wave Equation
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中文总结 AI 辅助
本文针对波动方程的时空伽辽金边界元法，提出一种分解凸多面体积分域的新积分方法，实现指数收敛，经散射问题测试验证有效性。
中文摘要 AI 辅助
延迟层势的时空伽辽金离散会得到一个线性方程组，其系数由试函数和检验函数单元上的积分表示。由于核在原点处奇异，且在双曲锥面上不连续，因此需要精心设计的求积格式。本文提出一种新的积分方法，该方法对应的格式随求积点数量呈指数收敛。其关键在于将积分域视为凸多面体，并将其分解为更简单多面体的凸包，这些多面体经参数化后，使得奇异性和不连续性仅出现在单个变量中。该方法针对分段常数单元实现，并在具有已知解析解的散射问题上进行了测试。
英文摘要
The space-time Galerkin discretization of retarded layer potentials leads to a linear system where the coefficients are expressed in terms of integrals over the ansatz and test elements. They require carefully designed quadrature schemes because the kernel is singular in the origin and is discontinuous across the hyperbolicity cone. This paper introduces a new integration approach that leads to a scheme that converges exponentially with the number of quadrature points. The key here is to consider the integration domain as a convex polytope and to devise a decomposition into the convex hulls of simpler polytopes which are parameterized such that the singularity and discontinuity occurs in a single variable. The method is implemented for piecewise constant elements and tested on a scattering problem with known analytic solution.
