The influence of the mass matrix on the dispersive nature of the semi-discrete, second-order wave equation

The application of discrete solution methods to the second-order wave equation can yield a dispersive representation of the non-dispersive wave propagation problem resulting in a phase speed that depends, not only upon the wavelength of the signal being propagated, but also upon the direction of propagation. In this work, the dependence of the dispersive errors on the wave propagation direction, mesh aspect ratio and wave number is investigated with the goal of understanding and hopefully reducing the phase and group errors associated with the two-dimensional bilinear finite element. An analysis of the dispersive effects associated with the consistent, row-sum lumped and higher-order mass matrices has led to a reduced-coupling 'penta-diagonal' mass matrix that yields improved phase and group errors with respect to wavelength and propagation direction. The influence of row-sum lumping the finite element mass matrix is demonstrated to always introduce lagging phase and group error, while a linear combination of the lumped and consistent mass matrices, i.e. a higher-order mass matrix, is shown to improve the dispersion characteristics of both the reduced and full-integration element. Using a 2.5% phase error metric, the higher-order mass matrix can reduce the mesh resolution requirements relative to the lumped mass matrix by nearly a factor of two in one dimension; a factor of 4 in two dimensions. Similarly, the reduced-coupling mass matrix can reduce the grid requirements by a factor of approximately four relative to the lumped mass matrix; a factor of 16 in two dimensions. Although the reduced-coupling mass matrix requires additional storage relative to the lumped mass, the factor of 16 reduction in grid requirements promises tremendous overall savings for computing a fixed wavelength spectrum, or alternatively for resolving shorter wavelengths with a fixed grid resolution. (C) 1999 Elsevier Science S.A. All rights reserved.