Reducing the low-wavenumber dispersion error by building the Lagrange dual problem with a powerful local restriction

Abstract

Abstract The broadband finite-difference (FD) coefficients computed by a cost function have been widely applied to suppress of numerical dispersion. Under the same conditions, the FD coefficients with small low-wavenumber dispersion error will produce a more accurate numerical solution in the long-time seismic wave simulation. Thus, how to reduce the low-wavenumber dispersion error becomes a crucial problem. According to the research into the zero-point position at the dispersion curve for three types of cost function, we found that the more zero points concentrate on the low-wavenumber region, the less the dispersion error. Therefore, the concentration of zero points is a good way to reduce dispersion error, which can be implemented by modified wavenumbers of zero points. Then, we design a Lagrange dual problem with a restriction based on the modified wavenumbers. The requirements for constructing the Lagrange dual problem are the optimization function and restricted condition, where the former is based on the dispersion relation, and the latter comprises the modified wavenumbers. The solution of this optimization problem, calculated by the dual ascent method, affords a less low-wavenumber dispersion error than the solution yielded by the alternating direction method of multipliers (ADMM). The theoretical analysis and numerical modeling suggest that the proposed method is superior to the existing optimal FD coefficients in reducing numerical error accumulation in low-frequency simulation.