Open AccessAn implicit staggered‐grid finite‐difference method for seismic modelling
Author(s) -
Liu Yang,
Sen Mrinal K.
Publication year - 2009
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.2009.04305.x
Subject(s) - tridiagonal matrix , computation , alternating direction implicit method , grid , mathematics , matrix (chemical analysis) , finite difference , order (exchange) , taylor series , tridiagonal matrix algorithm , finite difference method , mathematical analysis , algorithm , geometry , physics , eigenvalues and eigenvectors , materials science , finance , quantum mechanics , economics , composite material
SUMMARY We derive explicit and new implicit staggered‐grid finite‐difference (FD) formulas for derivatives of first order with any order of accuracy by a plane wave theory and Taylor's series expansion. Furthermore, we arrive at a practical algorithm such that the tridiagonal matrix equations are formed by the implicit FD formulas derived from the fractional expansion of derivatives. Our results demonstrate that the accuracy of a (2 N + 2)th‐order implicit formula is nearly equivalent to or greater than that of a (4 N )th‐order explicit formula. The new implicit method only involves solving tridiagonal matrix equations. We also demonstrate that a (2 N + 2)th‐order implicit formulation requires nearly the same amount of memory and computation as those of a (2 N + 4)th‐order explicit formulation but attains the accuracy achieved by a (4 N )th‐order explicit formulation when additional cost of visiting arrays is not considered. Our analysis of efficiency and numerical modelling results for elastic wave propagation demonstrates that a high‐order explicit staggered‐grid method can be replaced by an implicit staggered‐grid method of some order, which will increase the accuracy but not the computational cost.