Variable-order optimal implicit finite-difference schemes for explicit time-marching solutions to wave equations

Abstract

The time-space-domain finite-difference method (FDM) is widely used in forward modeling of wave equations. Conventional explicit FDMs (EFDMs) (with high order in space and the second order in time) would result in apparent temporal and spatial dispersion for high frequencies and large time steps. Moreover, the saturation effect of Taylor expansion seriously restricts the improvement of bandwidth coverage and efficiency of EFDMs. We have developed a variable-order optimization scheme of the implicit FDM (IFDM) to improve the computational efficiency and numerical accuracy of forward modeling. Then, we applied time-dispersion transforms (TDTs) to filter out the temporal dispersion generated by the second-order temporal approximations. Our method greatly alleviates the saturation effect of the high-order spatial finite-difference (FD) operators. Dispersion analysis indicates that the optimized coefficients of the IFDM based on the Remez algorithm can achieve the widest bandwidth (close to the Nyquist wavenumber), which corresponds to the shortest length of the spatial FD operator under a given error threshold. Our method has great potential to approach the highest spectral accuracy but with minimal increase in computational cost. Numerical experiments indicate that the combination of the variable-order optimization of IFDM and TDTs can significantly reduce numerical dispersion. Compared with traditional methods, our scheme is more advantageous for numerical simulation of large-scale geologic models because it has the least amount of calculation burden under the same accuracy requirements.