F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Efficient Long Time Computations of Time-domain Boundary Integrals for 2d and Dissipative Wave Equation Efficient Long Time Computations of Time-domain Boundary Integrals for 2d and Dissipative Wave Equation
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Efficient Long Time Computations of Time-domain Boundary Integrals for 2d and Dissipative Wave Equation Efficient Long Time Computations of Time-domain Boundary Integrals for 2d and Dissipative Wave Equation
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F ì Ur Mathematik in Den Naturwissenschaften Leipzig 二维和耗散波动方程时域边界积分的高效长时间计算 二维和耗散波动方程时域边界积分的高效长时间计算
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通讯作者：
Volker Gruhne
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作者：
L. Banjai;Volker Gruhne
均匀介质中的线性双曲偏微分方程，例如描述声波传播和散射的波动方程，可以重写为时域边界积分方程。当强惠更斯原理不成立时，我们提出了此类方程的数值离散化的有效实现。对于数值离散化，我们在时间上使用卷积求积，在空间上使用标准边界元方法。时间求积导致权重 W j 与在等间隔时间点评估的边界密度的离散卷积。如果强惠更斯原理成立，则对于足够大的 j，W j 会以指数方式快速收敛到 0。如果强惠更斯原理不成立，例如在均匀的空间维度或存在一些阻尼时，权重永远不会为零，从而给高效数值计算带来困难。在本文中，我们证明了卷积权重的核在某种意义上近似于时域基本解，并且如果两者在空间上微分，则同样成立。基本解的尾部非常平滑，这意味着权重的尾部是平滑的并且可以有效地进行插值。我们讨论了整个数值方案的有效实现并提出了数值实验。
Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be rewritten as a time-domain boundary integral equation. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens' principle does not hold. For the numerical discretization, we make use of convolution quadrature in time and standard boundary element method in space. The quadrature in time results in a discrete convolution of weights W j with the boundary density evaluated at equally spaced time points. If the strong Huygens' principle holds, W j converge to 0 exponentially quickly for large enough j. If the strong Huygens' principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation. In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.