On time integration of groundwater flow equations by spectral methods

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 ground water flow, direct integration methods are generally to be preferred.