Performance analysis and optimization of finite-difference schemes for wave propagation problems

In the present paper, we gauge the performance of finite-difference schemes with Runge–Kutta time integration for wave propagation problems by rigorously defining appropriate cost and error metrics in a simple setting represented by the linear advection equation. Optimal values of the grid spacing and of the time step are obtained as a result of a cost minimization (for given error level) procedure. The theory suggests superior performance of high-order schemes when highly accurate solutions are sought for, and in several space dimensions even more. The analysis of the global discretization error shows the occurrence of two (approximately independent) sources of error, associated with the space and time discretizations. The improvement of the performance of finite-difference schemes can then be achieved by trying to separately minimize the two contributions. General guidelines for the design of problem-tailored, optimized schemes are provided, suggesting that significant reductions of the computational cost are in principle possible. The application of the analysis to wave propagation problems in a two-dimensional environment demonstrates that the analysis carried out for the scalar case directly applies to the propagation of monochromatic sound waves. For problems of sound propagation involving disparate length-scales the analysis still provides useful insight for the optimal exploitation of computational resources; however, the actual advantage provided by optimized schemes is not as evident as in the single-scale, scalar case.