The relation between finite differences in time and the Chebyshev polynomial recursion

Abstract

In this paper, we show that the solution of the coupled wave equation system using a finite difference approximation in time follows exactly the recursion of the modified Chebyshev polynomials in AΔt where A is the first-order wave equation matrix system and Δt is the time step. The same connection is found with the solution of the first-order wave equation but with the Chebyshev polynomials in ΦΔt where Φ is a pseudodifferential operator. After that, we also show that the recursion for the modified Chebyshev polynomials which appears in the solution of the full wave equation using the rapid expansion method (REM) has the same connection with the finite difference method for the full wave equation using a second order time-difference approximation. Using the pseudospectral method for spatial derivatives and the connection between the Chebyshev polynomials and finite-difference approximation in time we can easily find the stability condition exploring only the range of validity of the Chebyshev polynomials. Thus, we obtain finite-difference schemes for the wave equation which are explicit and stable. The Chebyshev polynomial recursion can be used to march the wave field in time generating the stable propagation of seismic waves free of numerical dispersion. To validate the results obtained here, using the Chebyshev polynomial recursion, we compare the seismic responses from two synthetic model examples with the results obtained by the conventional REM solution.