Optimized explicit finite-difference schemes for spatial derivatives using maximum norm

Keywords: Optimized scheme Explicit finite-difference Numerical dispersion Maximum norm Simulated annealing algorithm a b s t r a c t Conventional explicit finite-difference methods have difficulties in handling high-frequency components due to strong numerical dispersions. One can reduce the numerical dispersions by optimizing the constant coefficients of the finite-difference operator. Different from traditional optimized schemes that use the 2-norm and the least squares, we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm. Both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects. First, it provides us with much more flexibility when designing the objective functions; thus we can use various possible forms and contents to make the objective functions more reasonable. Second, it allows for tighter error limitation, which is shown to be necessary to avoid rapid error accumulations for simulations on large-scale models with long travel times. Finally, it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits, which means greater savings in computational efforts and memory demand. The explicit finite-difference (FD) scheme is one of the most popular approaches used in various numerical simulations, because it is simple in its numerical implementation and is powerful in handling complex media. However, the conventional explicit FD method has serious numerical artifacts in the presence of high-frequency components and/or coarse grids. This problem would dramatically increase both the demands on the memory and computational cost, especially for large-scale models [7], since a fine grid should be properly designed and a high-order FD operator should be applied. A popular way to avoid this problem is to manually decrease the dominant frequency. This method could result in an acceptable running time, but would result in very limited spatial resolutions, because high-frequency components are necessary for improving the final resolutions. Another way is to apply advanced methods that have less numerical dispersion, such as optimized explicit FD methods and implicit FD methods (either conventional or optimized). Compared with implicit FD methods, explicit FD methods usually have much less computational cost. Therefore, we prefer to develop the optimized scheme of explicit FD methods to further reduce the numerical dispersions while maintaining the computational cost. Optimized schemes of FD methods appeared two decades ago [10,17]. It has been widely used to reduce the numerical …