基于非结构网格求解三维D'Alembert介质中声波方程的并行加权Runge-Kutta间断有限元方法

间断有限元方法(Discontinuous Galerkin method,简称DGM)在求解地震波动方程时具有低数值频散、网格剖分灵活等优点,因此,为适应数值模拟对模拟精度和复杂地质结构的要求,本文提出一种新的加权Runge-Kutta间断有限元(weighted Runge-Kutta discontinuous Galerkin,简称WRKDG)方法,用于求解三维D′Alembert介质中声波方程.本文不仅详细推导了其数值格式,特别地,根据常微分方程理论给出了满足数值稳定性条件的一般经验公式,并首次对该方法的数值频散和耗散进行了深入分析,且考虑了耗散参数对结果的影响.同时,我们也对该方法进行了精度测试,并分析了3D情形下WRKDG方法的并行加速比,结果表明3D WRKDG方法具有良好的并行性.最后,我们给出了包含均匀模型、非规则几何模型以及非均匀Marmousi模型在内的数值模拟算例.结果表明,该方法不仅计算准确,能与解析解很好地吻合,且能有效模拟包含球体在内的非规则模型及非均匀Marmousi模型中的衰减声波波场.数值模拟实验进一步验证了WRKDG方法在求解三维D′Alembert介质中声波方程时的正确性和有效性,并获得了对这种强衰减介质中波传播特征的规律性新认识.

A parallel weighted Runge-Kutta discontinuous galerkin method for solving acousitc wave equations in 3D D'Alembert media on unstructured meshes

Discontinuous Galerkin method(DGM)is a widely used numerical algorithm.It has the advantages of high accuracy,flexibility in dealing with boundary conditions,easy parallelism,and small numerical dispersion when solving seismic wave equations.In order to satisfy the numerical simulation for accuracy and complex geological structures,in this paper,we suggest a weighted Runge-Kutta discontinuous Galerkin(WRKDG)method for solving the acoustic wave equation in three-dimensional(3D)medium with strong attenuation—D′Alembert medium on unstructured meshes.The numerical scheme is derived in detail,and the general numerical stability conditions are presented based on the theory of ordinary differential equations.The numerical dispersion and dissipation of WRKDG method are also investigated for the first time,including the influence of dissipation parameters on the analysis results.In addition,we carry out a convergence test of this method,and analyze the parallel speedup ratio of the WRKDG method in 3D case.The results show that the 3D WRKDG method has good parallel capabilities.Finally,we present several numerical examples in complex media with strong attenuation,including an homogeneous model,an irregular geometric model,and the heterogeneous Marmousi model.The results show that the method is not only accurate and in good agreement with the analytical solution,but also can effectively simulate the acoustic wave field in irregular model including sphere and heterogeneous Marmousi model.Finally,we present several numerical examples.Numerical results further verify the correctness and effectiveness of the 3D WRKDG method in solving the scalar wave equation in D′Alembert medium,and they clearly show the wave propagation characteristics of this strong attenuation medium.