Introduction to the Distributional Finite Difference Method for 3D Seismic Wave Propagation and Comparison With the Spectral Element Method

We have extended the distributional finite difference method (DFDM) to simulate the seismic-wave propagation in 3D regional earth models. DFDM shares similarities to the discontinuous finite element method on a global scale and to the finite difference method locally. Instead of using linear staggered finite-difference operators, we employ DFDM operators based on B-splines and a definition of derivatives in the sense of distributions, to obtain accurate spatial derivatives. The weighted residuals method used in DFDM's locally weak formalism of spatial derivatives accurately and naturally accounts for the free surface, curvilinear meshing, and solid-fluid coupling, for which it only requires setting the shear modulus and the continuity condition to zero. The computational efficiency of DFDM is comparable to the spectral element method (SEM) due to the more accurate mass matrix and a new band-diagonal mass factorization. Numerical examples show that the superior accuracy of the band-diagonal mass and stiffness matrices in DFDM enables fewer points per wavelength, approaching the spectral limit, and compensates for the increased computational burden due to four Lebedev staggered grids. Specifically, DFDM needs 2.5 points per wavelength, compared to the five points per wavelength required in SEM for 0.5% waveform error in a homogeneous model. Notably, while maintaining the same accuracy in the solid domain, DFDM demonstrates superior accuracy in the fluid domain compared to SEM. To validate its accuracy and flexibility, we present various 3D benchmarks involving homogeneous and heterogeneous regional elastic models and solid-fluid coupling in both Cartesian and spherical settings. Numerical simulation of the wave equation is an important tool for exploring the internal properties of objects. However, high efficiency and high precision in numerical simulations often trade off against each other. The recently proposed distributional finite difference method appears to remedy such trade-offs to some extent. Using the numerical simulation of the seismic wave equation in 3D regional models as an example, we demonstrate the high efficiency and high precision of the new distributional finite difference method through a detailed comparison with the popular spectral element method. This illustrates its potential for applications in both global seismology and exploration geophysics. We implement the distributional finite difference method (DFDM) for 3D seismic wave propagation DFDM shows promise for improving accuracy, flexibility, and efficiency against the Spectral Element Method DFDM's accurate mass and stiffness matrices enable approximate to 2.5 points per wavelength, reducing computational burden with Lebedev staggered grids