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

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.