Accurate time domain computation of linear wave propagation using Chebyshev collocation and matrix diagonalization

Abstract

Accurate time domain computation of the wave propagation phenomena is of great interest to many fields of studies. The spectral method of Chebyshev collocation is adopted for smooth problems of linear waves. Space differentiation is discretized by Chebyshev derivative matrices and the wave equation is cast as a second-order ordinary differential equation in time domain. The general boundary conditions involving pressure and acoustic particle velocity are discretized and absorbed into the system equations for the unknown vector containing all grid points. The system equations are solved accurately by the method of matrix diagonalization, which involves the finding of the eigenvalues and eigenvectors for the appropriate combinations of the damping and stiffness matrices. For this solution, the time step for stable and accurate computation is only limited by the consideration of round-off errors when the exponential functions are evaluated. Comparison is made with analytical solutions in the one-dimensional case, and good agreement is obtained. The only perceived drawback would be the size of matrices that can be alleviated by the use of domain decomposition approaches. The physical meaning of the eigenvalues derived from the discretization procedure is discussed.