A new method for implicit time integration of the acoustic wave equation

Abstract

Wave equations are usually simulated using explicit time integration methods. But the strong restriction on time step sizes in media with large differences in wave speeds can lead to high computational cost. Implicit time integration methods on the other hand can achieve better stability properties, which allow larger time steps. In this work, we consider diagonally implicit Runge Kutta-based (DIRK) time integration schemes for second-order ordinary differential equations (ODEs) applied to the acoustic wave equation and attempt to identify efficient combinations of time integration schemes, Runge-Kutta coefficients, and linear system solver parameters. We derive new general time integration schemes for second-order ODEs similar to the diagonally implicit Runge-Kutta-Nyström (DIRKN) scheme. But instead of reordering the DIRK equations to solve for the second temporal derivative of the acoustic pressure, they solve for the first derivative (DIRK-2-1) or the variable itself (DIRK-2-0). The latter scheme allows further substitutions when applied to linear ODEs like the acoustic wave equation. These substitutions lead to another scheme (DIRK-2-0s) that avoids matrix-vector multiplications and therefore has potential cost benefits. Because DIRK-2-1 has higher costs than the others, we do not include it in the performance analysis. We apply these time integration schemes to the acoustic wave equation and analyze their performance. To solve the arising linear systems, the conjugate gradient method with a multigrid preconditioner is used. We analyze the efficiency of the schemes with the coefficients of various highly-stable DIRK and DIRKN methods and for various configurations of the linear system solver. The best configurations for the three schemes DIRKN, DIRK-2-0, and DIRK-2-0s perform similarly in regard to cost and accuracy. The new schemes do not outperform DIRKN consistently. While avoidance of the matrix-vector multiplications decreases the costs per time step in theory, DIRK-2-0s amplifies the error of inexact linear system solvers more than the alternatives. Therefore, it requires lower tolerances for the conjugate gradient method which increases its overall cost. While their performance is similar, DIRKN tends to deliver the most efficient results in a medium accuracy range, which is probably relevant for most applications. Regarding the choice of methods (sets of coefficients), a general recommendation is not possible. At different levels of accuracy, different methods are the most efficient. The effects of varying solver parameters like the conjugate gradient method’s tolerance and the use of the Galerkin condition are also difficult to predict. Therefore, efficient combinations of a time integration scheme, Runge-Kutta coefficients, and linear system solver parameters should be determined experimentally for the desired level of accuracy. The numerical experiments show that our approach is unlikely to outperform efficient explicit methods. While there are combinations that lead to lower costs than some explicit methods, their accuracy is also reduced. Increasing the time step size for implicit methods relative to explicit ones reduces their cost further but also makes them less accurate, even for relatively high temporal resolutions of the simulated waves. This shows that the beneficial effect of small time steps on accuracy should not be underestimated.