Simplified implicit finite-difference method of spatial derivative using explicit schemes with optimized constant coefficients based on lt;igt; Llt;/igt;lt;subgt;1lt;/subgt; norm

Abstract

The explicit finite-difference (FD) method is widely used in numerical simulation of seismic wave propagation to approximate spatial derivatives. However, both the traditional and optimized high-order explicit FD methods suffer from the saturation effect, which seriously restricts the improvement of numerical accuracy. In contrast, the implicit FD method approximates the spatial derivatives in the form of rational functions and thus can obtain much higher numerical accuracy with relatively low orders; however, its computational cost is expensive due to the need to invert a multi-diagonal matrix. We derive an explicit strategy for the implicit FD method to reduce the computational cost, constructing the implicit FD method with the discrete Fourier matrix; then, we transform the inversion of the multi-diagonal matrix into an explicit matrix multiplication; next, we construct an objective function based on the L1 norm to reduce approximation error of the implicit FD method. The proposed explicit strategy of the implicit FD method can avoid inverting the multi-diagonal matrix, thus improving the computational efficiency. The proposed constant coefficient optimization method reduces the approximation error in the medium-wavenumber range at the cost of tolerable deviation (smaller than 0.0001) in the low-wavenumber range. For the 2D Marmousi model, the root-mean-square error of the numerical results obtained by the proposed method is one-fifth that of the traditional implicit FD method with the same order (i.e., 5/3) and one-third that of the traditional explicit FD method with much higher orders (i.e., 72). The significant reduction of numerical error makes the proposed method promising for numerical simulation in large-scale models, especially for long-time simulations.