Optimal blended spectral-element operators for acoustic wave modeling

Abstract

Spectral-element methods, based on high-order polynomials, are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media. However, to retain computational efficiency, very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided. In the present work, we devise an approach for reducing such errors by considering modified discrete wave operators. We analyze consistent and lumped operators together with blended operators (weighted averages of consistent and lumped operators). Furthermore, using the operator-blending approach and a novel dispersion analysis method, we develop optimal spectral-element operators that have increased numerical accuracy, without resorting to very high order operators. The new operators are faster and computationally more efficient than consistent operators. Our approach is based on the tensor product decomposition of the element matrices into 1D factors. We apply standard lumping to the factor associated with the 1D mass matrix. A simplified numerical dispersion analysis of arbitrary order and spatial dimension provides a practical criterion for weighting consistent and lumped matrices. The approach is general and is valid for solving both the time-dependent and the stationary (Helmholtz) wave equations.