Reducing error accumulation of optimized finite-difference scheme using the minimum norm

Abstract

The finite-difference (FD) scheme is popular in the field of seismic exploration for numerical simulation of wave propagation; however, its accuracy and computational efficiency are restricted by the numerical dispersion caused by numerical discretization of spatial partial derivatives using coarse grids. The constant-coefficient optimization method is used widely for suppressing the numerical dispersion by tuning the FD weights. Although gaining a wider effective bandwidth under a given error tolerance, this method undoubtedly encounters larger errors at low wavenumbers and accumulates significant errors. We have developed an approach to reduce the error accumulation. First, we construct an objective function based on the [Formula: see text] norm, which can constrain the total error better than the [Formula: see text] and [Formula: see text] norms. Second, we translated our objective function into a constrained [Formula: see text]-norm minimization model, which can be solved by the alternating direction method of multipliers. Finally, we perform theoretical analyses and numerical experiments to illustrate the accuracy improvement. The proposed method is shown to be superior to the existing constant-coefficient optimization methods at the low-wavenumber region; thus, we can obtain higher accuracy with less error accumulation, particularly at longer simulation times. The widely used objective functions, defined by the [Formula: see text] and [Formula: see text] norms, could handle a relatively wider range of accurate wavenumbers, compared with our objective function defined by the [Formula: see text] norm, but their actual errors would be much larger than the given error tolerance at some azimuths rather than axis directions (e.g., about twice at 45°), which greatly degrade the overall numerical accuracy. In contrast, our scheme can obtain a relatively even 2D error distribution at various azimuths, with an apparently smaller error. The peak error of the proposed method is only 40%–65% that of the [Formula: see text] norm under the same error tolerance, or only 60%–80% that of the [Formula: see text] norm under the same effective bandwidth.