An Explicit Solution Suited for HPC to Calculate the Implicit FDs
H. Liu, Hussain Salim · 2022
Summary The finite difference solution of the second-order acoustic wave equation is a fundamental algorithm in seismic exploration. In contrast to the explicit finite difference (EFD) schemes that usually suffer from the so-called “saturation effect”, the implicit FD schemes can obtain a spectral-like resolution with a relatively short operator length. Unfortunately, these implicit schemes are not widely implemented because band matrices need to be solved implicitly, which is not suitable for high-performance computing (HPC). We introduce an explicit solution to overcome this limitation by applying causal and anti-causal integrations, and the new solution can be proved to be equivalent to the traditional implicit LU decomposition solution. Besides, we also compare the accuracy of the new schemes with traditional EFD schemes up to the 32nd order, and the results indicate that the optimized implicit FD scheme is more accurate. The computational cost for the newly proposed scheme is standard 8th order EFD plus two causal and anti-causal integrations, which can be realized recursively.