Open AccessNumerical modelling method for wave propagation in a linear viscoelastic medium with singular memory
Author(s) -
Lu JianFei,
Hanyga Andrzej
Publication year - 2004
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.2004.02409.x
Subject(s) - mathematics , viscoelasticity , relaxation (psychology) , ordinary differential equation , mathematical analysis , spectral method , fractional calculus , euler method , discretization , euler equations , numerical analysis , differential equation , fourier transform , partial differential equation , physics , psychology , social psychology , thermodynamics
SUMMARY A numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory is developed in this paper. For a demonstration of the method, the Cole–Cole model of viscoelastic relaxation is adopted here. A formulation of the Cole–Cole model based on internal variables satisfying fractional relaxation equations is applied. In order to avoid integrating and storing of the entire history of the variables, a new method for solving fractional differential equations of arbitrary order based on a set of secondary internal variables is developed. Using the new method, the velocity–stress equations and the fractional relaxation equations are reduced to a system of first‐order ordinary differential equations for the velocities, stresses, primary internal variables as well as the secondary internal variables. The horizontal spatial derivatives involved in the governing equations are calculated by the Fourier pseudo‐spectral (PS) method, while the vertical ones are calculated by the Chebychev PS method. The physical boundary conditions and the non‐reflecting conditions for the Chebychev PS method are also discussed. The global solution of the first‐order system of ordinary differential equations is advanced in time by the Euler predictor–corrector methods. For the demonstration of our method, some numerical results are presented.