Application of a~spectral method for numerical modeling of propagation of seismic waves in porous media for dissipative case

Kh. Kh. Imomnazarov ; A. A. Mikhailov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 17 (2014) no. 2, pp. 139-147

Résumé

This paper presents the algorithm, based on the application of the spectral Laguerre method for approximation of temporal derivatives as applied to the problem of seismic wave propagation in the porous media with dissipation of energy. The initial system of equations is written down as the first order hyperbolic system in terms of velocities, stresses and pore pressure. For the numerical solution of the problem in question, the method of a combination of the analytical Laguerre transformation and a finite difference method is used. The proposed method of the solution can be considered to be an analog to the known spectral method based on the Fourier transform. However, unlike the Fourier transform, application of the integral Laguerre transform with respect to time allows us to reduce the initial problem to solving a system of equations in which the parameter of division is present only in the right-hand side of equations and has a recurrent dependence. As compared to the time-domain method, with the help of an analytical transformation in the spectral method it is possible to reduce an original problem to solving a system of differential equations, in which there are only derivatives with respect to spatial coordinates. This allows us to apply a known stable difference scheme for recurrent solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Thus, because of the presence of the second longitudinal wave with a low velocity, the use of difference schemes in all coordinates for stable solutions requires a consistent small step both with respect to time and space, which inevitably results in an increase in computer costs.