A new iterative solver for the time-harmonic wave equation

Abstract

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

“…This preconditioner is a member of the family of shifted Laplacian operators, introduced in [20], and its inverse can be efficiently approximated by means of a multigrid iteration. Two-dimensional results, representative for geophysical applications, generated by second-order finite differences, have been presented in [26] and 3D results in [27].In this paper, we generalize this solver and include, in particular, a fourth-order discretization of the Helmholtz operator in our discussion. The multigrid preconditioner is enhanced, in the sense that we replace the point-wise Jacobi smoother in the multigrid preconditioner by a variant of the incomplete lower-upper factorization smoother, ILU(0).…”

“…This preconditioner is a member of the family of shifted Laplacian operators, introduced in [20], and its inverse can be efficiently approximated by means of a multigrid iteration. Two-dimensional results, representative for geophysical applications, generated by second-order finite differences, have been presented in [26] and 3D results in [27].…”

High-performance finite-element simulations of seismic wave propagation in three-dimensional nonlinear inelastic geological media

“…The frequency domain approach is also used in geophysics, especially for inverse problems in the acoustic case (e.g. [11], [12]), but far more often the problem is solved in time (for most direct problems and also for inverse problems in the more complex elastic, viscoelastic or poroelastic cases (see e.g. [13], [14]).…”

Seismic Waves - Research and Analysis

“…Iterative solvers provide an alternative approach for solving the time-harmonic wave equation (Erlangga & Herrmann, 2008;Plessix, 2007;Riyanti et al, 2006;. Iterative solvers are currently implemented with Krylov-subspace methods (Saad, 2003) that are preconditioned by the solution of the dampened time-harmonic wave equation.…”