三维波动方程时空域混合网格有限差分数值模拟方法

Mixed-grid finite-difference method for numerical simulation of 3D wave equation in the time-space domain

常规高阶和时空域高阶有限差分方法广泛应用于三维标量波动方程的数值模拟,这两种差分方法仅利用笛卡尔坐标系中的坐标轴网格点构建三维Laplace差分算子,相应的差分离散波动方程本质上仅具有2阶差分精度,模拟精度低.本文将三维笛卡尔坐标系中非坐标轴网格点分为两类:坐标平面内的非坐标轴网格点和坐标平面外的非坐标轴网格点,系统推导出了两类非坐标轴网格点构建三维Laplace差分算子的方法,进而提出了一种利用坐标轴网格点和非坐标轴网格点共同构建三维Laplace差分算子的混合网格有限差分方法,并利用时空域频散关系和泰勒展开建立差分系数方程,推导出了差分系数的通解.相比常规高阶和时空域高阶差分格式的2阶差分精度,时空域混合网格差分离散波动方程理论上能够达到任意偶数阶差分精度,模拟精度显著提高,同时稳定性更强.频散分析表明:相比常规高阶和时空域高阶差分格式,在计算效率基本相同时,时空域混合网格差分格式能更有效地减小数值频散,减弱数值各向异性,模拟精度更高;在模拟精度基本相当时,混合网格差分格式能采用更大的时间采样间隔,计算效率更高.数值模拟实例进一步验证了混合网格差分格式在提高模拟精度和计算效率方面的先进性,也验证了其普遍适用性.

Conventional high-order and time-space domain high-order Finite-Difference(FD)methods have been widely used for 3D wave equation numerical simulation.These two FD schemes adopt the same FD stencil which only uses the axial grid nodes to construct the 3D Laplace difference operator,and the corresponding discrete difference wave equation can only reach 2nd-order accuracy.In this paper,we classify the off-axial grid nodes in the 3D Cartesian coordinate system into two categories:off-axial grid nodes in the coordinate plane and off-axial grid nodes outside the coordinate plane,and systematically derive the methods to construct the 3D Laplace difference operator with these two categories of off-axial grid nodes.Then a new kind of 3D mixed-grid FD scheme is proposed,which adopts a novel FD stencil combining the axial nodes and off-axial nodes to construct the Laplace difference operator and the FD coefficients are calculated based on the time-space domain dispersion relationship and Taylor expansion.The discrete difference wave equation derived from this new mixed-grid FD scheme can reach 4th-order,6th-order and arbitrary even-order difference accuracy.So it can significantly improve the modeling accuracy comparing to conventional high-order and time-space domain high-order FD schemes,and also has better stability.Dispersion analysis shows comparing to conventional high-order and time-space domain high-order FD schemes,the mixed-grid FD scheme can more effectively suppress the numerical dispersion and weaken the numerical anisotropy to obtain higher modeling accuracy with almost the same computational efficiency,and it can obtain higher computational efficiency by adopting larger time interval with almost the same modeling accuracy.Numerical modeling experiments further verify the superiority of the mixed-grid FD scheme in improving the modeling accuracy and computational efficiency,and also demonstrate its universal applicability.