Stability and Numerical Dispersion Analysis of Finite-difference Method for the Diffusive-viscous Wave Equation

The diffusive-viscous wave equation plays an important role in seismic exploration and it can be used to explain the frequency-dependent reflections observed both in laboratory and field data. The numerical solution to this type of wave equation is needed in practical applications because it is difficult to obtain the analytical solution in complex media. Finite-difference method (FDM) is the most common used in numerical modeling, yet the numerical dispersion relation and stability condition remain to be solved for the diffusive-viscous wave equation in FDM. In this paper, we perform an analysis for the numerical dispersion and Von Neumann stability criteria of the diffusive-viscous wave equation for second order FD scheme. New results are compared with the results of acoustic case. Analysis reveals that the numerical dispersion is inversely proportional to the number of grid points per wavelength for both cases of diffusive-viscous waves and acoustic waves, but the numerical dispersion of the diffusive-viscous waves is smaller than that of acoustic waves with the same time and spatial steps due to its more restrictive stability condition, and it requires a smaller time step for the diffusive-viscous wave equation than acoustic case.