Optimal staggered-grid finite-difference schemes based on the minimax approximation method with the Remez algorithm

Finite-difference (FD) schemes, especially staggered-grid FD (SFD) schemes, have been widely implemented for wave extrapolation in numerical modeling, whereas the conventional approach to compute the SFD coefficients is based on the Taylor-series expansion (TE) method, which leads to unignorable great errors at large wavenumbers in the solution of wave equations. We have developed new optimal explicit SFD (ESFD) and implicit SFD (ISFD) schemes based on the minimax approximation (MA) method with a Remez algorithm to enhance the numerical modeling accuracy. Starting from the wavenumber dispersion relations, we derived the optimal ESFD and ISFD coefficients by using the MA method to construct the objective functions, and solve the objective functions with the Remez algorithm. We adopt the MA-based ESFD and ISFD coefficients to solve the spatial derivatives of the elastic-wave equations and perform numerical modeling. Numerical analyses indicated that the MA-based ESFD and ISFD schemes can overcome the disadvantages of conventional methods by improving the numerical accuracy at large wavenumbers. Numerical modeling examples determined that under the same discretizations, the MA-based ESFD and ISFD schemes lead to greater accuracy compared with the corresponding conventional ESFD or ISFD scheme, whereas under the same numerical precision, the shorter operator length can be adopted for the MA-based ESFD and ISFD schemes, so that the computation time is further decreased.

On the Remes Algorithm for Rational Approximations

This paper is concerned with the minimax approximation of a discrete data set by rational functions. The second algorithm of Remes is applied. A crucial stage of this algorithm is solving the nonlinear system of leveling equations. In this paper, we will give a new approach for this purpose. In this approach, no initial guesses are required. Illustrative numerical example is presented.