A Weighted Runge-Kutta Method with Weak Numerical Dispersion for Solving Wave Equations (Communications in Computational Physics, 2010)
In this paper, we propose a weighted Runge-Kutta (WRK) method to solvethe 2D acoustic and elastic wave equations. This method successfully suppresses thenumerical dispersion resulted from discretizing the wave equations. In this method,the partial differential wave equation is first transformed into a system of ordinarydifferential equations (ODEs), then a third-order Runge-Kutta method is proposedto solve the ODEs. Like the conventional third-order RK scheme, this new methodincludes three stages. By introducing a weight to estimate the displacement and itsgradients in every stage, we obtain a weighted RK (WRK) method. In this paper, weinvestigate the theoretical properties of the WRK method, including the stability criteria,numerical error, and the numerical dispersion in solving the 1D and 2D scalarwave equations. We also compare it against other methods such as the high-ordercompact or so-called Lax-Wendroff correction (LWC) and the staggered-grid schemes.To validate the efficiency and accuracy of the method, we simulate wave fields in the2D homogeneous transversely isotropic and heterogeneous isotropic media. We concludethat the WRK method can effectively suppress numerical dispersions and sourcenoises caused in using coarse grids and can further improve the original RK methodin terms of the numerical dispersion and stability condition.