Application of the Lanczos Algorithm to the Solution of the Groundwater Flow Equation

The solution to the finite element matrix-differential equations resulting from the discretization of the groundwater flow equation is normally carried out by a finite difference approximation to the time derivative. The total computational effort in solving a fluid flow problem is then directly related to the number of unknowns and the number of time steps required to obtain accurate and stable solutions. The Lanczos algorithm uses an orthogonal matrix transformation to reduce the finite element equations to a much smaller tridiagonal system of first-order differential equations. This new system can be solved by astandard tridiagonal solution algorithm with very little computational effort. A matrix-vector multiplicationis then used to obtain the original solution at desired time steps. The algorithm is used to accurately simulate the drawdown of synthetic two-dimensional aquifers, including ones with substantial hydraulic conductivity contrasts. Th e method affords an efficient means of solving large problems,particularly when time durations are long