A matrix-transform numerical solver for fractional Laplacian viscoacoustic wave equation

Abstract

We have developed a matrix-transform method (MTM) to numerically solve the fractional Laplacian constant-[Formula: see text] viscoacoustic wave equation. The new method is based on a matrix representation of the fractional Laplacians, and it is different from the traditional Fourier-spectrum representation. With the MTM approach, the viscoacoustic wave equation is discretized in the space domain; thus, the periodic boundary caused by the Fourier-transform method can be avoided. Spatial discretization offers great convenience for handling various boundary conditions. In the MTM scheme, the fractional power of a matrix needs to be computed, and it can be realized with the help of eigenvalue decomposition (EVD). However, the application of EVD to a huge matrix is uneconomical and impractical. To avoid forming any huge matrix explicitly, we adopted the Lanczos approximation to compute the matrix-vector product directly. The key step in the Lanczos approximation is the involved Lanczos decomposition, which is implemented by an iterative flow. We determine how to choose the iteration number in the Lanczos decomposition to balance the wavefield simulation accuracy and the computational cost with the help of numerical examples. We also analyze the stability condition of MTM numerically, and we compare its computational cost with that of the traditional Fourier-transform method.