On time integration of groundwater flow equations by spectral methods

Abstract

A study of the computational performance of the recently introduced spectral approach for the time integration of finite element flow equations is performed. Two reduction methods are presented and discussed. One method relies on the classical modal decomposition wherein the leftmost eigenpairs of a generalized symmetric eigenvalue problem are evaluated one at a time by a cost-effective orthogonal deflation technique. The other uses a set of Lanczos vectors for the coordinate transformation matrix and involves a recursive product between the inverse of the stiffness matrix and a vector. An extensive analysis of the behavior of the two methods is made with a representative sample problem of subsurface flow solved over a regular and an irregular mesh. The results show that the spectral approach cannot be competitive from a computational viewpoint with a more traditional time-stepping or direct integration scheme (e.g., the Crank-Nicolson scheme). The eigenvector technique is slow to converge, and a large portion, if not all, of the "flow modes" need to be determined to accurately describe the desired solution. The Lanczos algorithm may converge faster in the Lanczos vector space. However, its actual performance depends to a large extent on the initial guessed vector and is adversely influenced by the requirement to solve an equivalent steady state problem at each iteration. It is concluded that for a robust and efficient finite element code of transient groundwater flow, direct integration methods are generally to be preferred.

Comment on “On time integration of ground water flow equations by spectral methods” by G. Gambolati

“…Dunbar andWoodbury [1987, 1989] extend the approach and present examples in aquifer modeling. The essence of the approach is also described by Gambolati [1993]. For completeness and to introduce notation, a brief description of the method is given here.…”

“…Another of Efficiency of the Lanczos Algorithm Gambolati [1993] concludes that the direct time integration of (1) by standard methods is more efficient than the Lanczos method. An operation count comparison between direct integration of (1) by the Crank-Nicolson method (DICN) and integration using the Lanczos reduction method is shown in Figure 5, which shows a plot of floating point operation (flop) counts (in millions) versus time step for both the DICN and Lanczos methods.…”

“…It was used by both DW (1989) and Gambolati [1993] Gambolati [1993], it is necessary to assume that the flow rate given in Table 1 Number of elements 800 ( …”

“…It was used by Gambolati [1993] to compare the convergence of the Lanczos method with that of direct integration of (1) by the Crank-Nicolson method. Figure 3 shows the finite element mesh used.…”

Parallel eigenanalysis of multiaquifer systems

“…. , u N ] is the modal matrix [3]. Only a number q of the leftmost eigenpairs need to be evaluated, since the importance of the flow modes, in capturing the actual response, decreases from smaller (leftmost) eigenvalues to larger (rightmost) ones.…”