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An Explicit Method Based on the Implicit Runge–Kutta Algorithm for Solving Wave Equations

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China dhyang{at}math.tsinghua.edu.cn

A new explicit differentiator series method based on the implicit Runge–Kutta method, called the IRK-DSM in brief, is developed for solving wave equations. To develop the new algorithm, we first transform the wave equation, usually described as a partial differential equation (PDE), into a system of first-order ordinary differential equations (ODEs) with respect to time t. Then we use a truncated differentiator series method of the implicit Runge–Kutta method to solve the semidiscrete ordinary differential equations, while the high-order spatial derivatives included in the ODEs are approximated by the local interpolation method. We analyze the theoretical properties of the IRK-DSM, including the stability criteria for solving the 1D and 2D acoustic-wave equations, numerical dispersion, discretizing error, and computational efficiency when using the IRK-DSM to model acoustic-wave fields. For comparison, we also present the stability criteria and numerical dispersion of the so-called Lax–Wendroff correction (LWC) methods with the fourth-order and eighth-order accuracies for the 1D case. Promising numerical results show that the IRK-DSM provides a useful tool for large-scale practical problems because it can effectively suppress numerical dispersions and source-noises caused by discretizing the acoustic- and elastic-wave equations when too-coarse grids are used or the models have a large velocity contrast between adjacent layers. Theoretical analysis and numerical modeling also demonstrate that the IRK-DSM, through combining both the implicit Runge–Kutta scheme with good stability condition and the approximate differentiator series method, is a robust wave-field modeling method.