伯恩斯坦-贝塞尔加权调整不连续伽辽金方法用于异质介质中的波传播
Bernstein-Bezier weight-adjusted discontinuous Galerkin methods for wave propagation in heterogeneous media
AI总结：
本文提出了一种高效的不连续伽辽金方法，用于模拟具有子单元变化的异质介质中的波传播。该方法基于加权调整不连续伽辽金方法（WADG），能够实现任意异质介质的高阶精度。然而，WADG的计算成本随着近似阶数的增加而迅速增长。为此，本文提出了一种伯恩斯坦-贝塞尔加权调整不连续伽辽金方法（BBWADG），通过用固定次数多项式近似子单元异质性，将WADG的主要步骤表示为多项式乘法和$ L^2 $投影，并利用快速伯恩斯坦算法进行计算。该方法将总体计算复杂度从$ O(N^{2d}) $降低到$ O(N^{d+1}) $。数值实验展示了所提出方法的准确性，并且对BBWADG的GPU实现计算实验验证了该理论复杂度的实际实现。
摘要：
本文提出了一种高效的不连续伽辽金方法，用于模拟具有子单元变化的异质介质中的波传播。该方法基于加权调整不连续伽辽金方法（WADG），能够实现任意异质介质的高阶精度。然而，WADG的计算成本随着近似阶数的增加而迅速增长。为此，本文提出了一种伯恩斯坦-贝塞尔加权调整不连续伽辽金方法（BBWADG），通过用固定次数多项式近似子单元异质性，将WADG的主要步骤表示为多项式乘法和$ L^2 $投影，并利用快速伯恩斯坦算法进行计算。该方法将总体计算复杂度从$ O(N^{2d}) $降低到$ O(N^{d+1}) $。数值实验展示了所提出方法的准确性，并且对BBWADG的GPU实现计算实验验证了该理论复杂度的实际实现。
英文摘要：
This paper presents an efficient discontinuous Galerkin method to simulate wave propagation in heterogeneous media with sub-cell variations. This method is based on a weight-adjusted discontinuous Galerkin method (WADG), which achieves high order accuracy for arbitrary heterogeneous media. However, the computational cost of WADG grows rapidly with the order of approximation. In this work, we propose a Bernstein-Bézier weight-adjusted discontinuous Galerkin method (BBWADG) to address this cost. By approximating sub-cell heterogeneities by a fixed degree polynomial, the main steps of WADG can be expressed as polynomial multiplication and $L^2$ projection, which we carry out using fast Bernstein algorithms. The proposed approach reduces the overall computational complexity from $O(N^{2d})$ to $O(N^{d+1})$ in $d$ dimensions. Numerical experiments illustrate the accuracy of the proposed approach, and computational experiments for a GPU implementation of BBWADG verify that this theoretical complexity is achieved in practice.
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