In this work, we couple several time integration schemes to a finite-difference staggered-grid fourth-order accurate method for 1 -D wave propagation. These implementations include two conventional Runge Kutta (RK) schemes with third and fourth order accuracy, and two optimized RK methods for dissipation and dispersion reduction that also offer fourth order accuracy on linear problems. In addition, we also apply the time-staggering Leap frog integration with second and fourth order accuracy. As a nearly analytic reference, an exponentially time differencing scheme is also employed. We first carry out a eigenvalue stability analysis of all these methods to find limiting integration steps. Then, we quantify their dissipation and dispersion errors on solving a homogeneous problem at different grid resolutions and propagation distances. For all tests, accuracy of numerical results significantly increases with the discretization order, and the high precision of one member of the optimized fourth-order RK schemes is also accompanied by a large stability bound. Thus, we recommend to analyze this scheme family on heterogeneous and/or multidimensional problems.


Article metrics loading...

Loading full text...

Full text loading...


  1. Al-MohyA.H. and Higham, N.J.
    [2010] A New Scaling and Squaring Algorithm for the Matrix Exponential. SIAM Journal on Matrix Analysis and Applications, 31(3), 970–989.
    [Google Scholar]
  2. Blanch, J.O. and Robertsson, J.O.A.
    [1997] A modified Lax-Wendroff correction for wave propagation in media described by Zener elements. Geophysical Journal International, 131(2), 381–386.
    [Google Scholar]
  3. Bogey, C. and Bailly, C.
    [2004] A family of low dispersive and low dissipative explicit schemes for flow and noise computations. Journal of Computational Physics, 194(1), 194–214.
    [Google Scholar]
  4. Durran, D.R.
    [1999] Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. 1.
    [Google Scholar]
  5. Moczo, P., Kristek, J., Galis, M., Pazak, P. and Balazovjech, M.
    [2007] The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion. Acta Physica Slovaca, 52(2), 177–406.
    [Google Scholar]
  6. Rojas, O., Dunham, E., Day, S.M., Dalguer, L.A. and Castillo, J.E.
    [2009] Finite difference modeling of rupture propagation with strong velocity-weakening friction. Geophysical Journal International, 179, 1831–1858.
    [Google Scholar]
  7. Tselios, K. and Simos, T.
    [2005] Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. Journal of Computational and Applied Mathematics, 175(1), 173–181. Selected Papers of the International Conference on Computational Methods in Sciences and Engineering.
    [Google Scholar]
  8. Wicker, L.J. and Skamarock, W.C.
    [2002] Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097.
    [Google Scholar]

Data & Media loading...

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error