Lockheed Palo Alto Research Laboratory, Palo Alto, California. The solution to the finite ...... hand side of the domain from x equals 0 to 130 m (first 11 nodes at top). Material properties ..... 10, AGU, Washington, D.C., 1984. Jennings, A., Matrix ...
WATER
RESOURCES
RESEARCH,
VOL. 26, NO. 10, PAGES 2579-2590, OCTOBER
1990
Application of the Arnoldi Algorithm to the Solution of the Advection-Dispersion Equation ALLAN
D. WOODBURY
Department of Geological Engineering, University of Manitoba, Winnipeg, Manitoba, Canada W.
SCOTT DUNBAR
Acres International, Ltd., Niagara Falls, Ontario, Canada
BAHRAM
NOUR-OMID
Lockheed Palo Alto Research Laboratory, Palo Alto, California The solution to the finite element matrix differential equationsresultingfrom the discretization of the contaminant transport equation is normally carried out by a finite difference approximation to the time derivative. The total computational effort in simulatinga contaminant plume is then directly related to the number of unknowns and the number of time stepsrequired to obtain accurate and stable solutions. An alternative method is the Arnoldi algorithm which uses orthogonal matrix transformations to reduce the finite element equations to a much smaller upper Hessenberg system of first-order differential equations. This new system can be solved by a standard Crank-Nicolson algorithm with very little computational effort. A matrix-vector multiplication is then used to obtain the original solutionat desiredtime steps.The algorithmis usedto simulateaccurately the contaminantplumesfor a strip sourceareal aquifer, a cross-sectionalproblem, and the Borden landfill in Ontario, Canada. The Arnoldi algorithm showsan impressive613% increasein speedover the conventional Crank-Nicolson scheme for this latter case. The method affords an efficient means of solving large problems, particularly when time durations are long.
INTRODUCTION
A considerable
amount
of recent
research
has been de-
voted to finding alternate methods of solving the advectiondispersion equation. Recent developments are summarized in standard texts, including the work of Huyakorn and Pinder [1983], Javandel et al. [1984], de Marsily [1986], and Bear and Verruijt [1987]. These methods include (example papers indicated) the method of characteristics [Konikow and Bredehoeft, 1974, 1978], random walk particle models [Prickett et al., 1981], boundary elements [Brebbia and $kerit, 1984], multigrids [McCormick, 1989], and movinggrid coordinate systems [Farmer, 1986]. Hybrid numerical approaches such as principal direction and alternating direction Galerkin (ADG) have been recently promoted [e.g., Daus and Frind, 1985]. In spite of these advances, the finite element technique is the favored method of solution, particularly for geometrically irregular domains. However, it is well known that for complex systems in two or three dimensionsthe demand on computingresourcescan be high because of the need to store large matrices within a time marching algorithm and to adhere to rigorousgrid Peclet and Courant number criteria [Carey and Sepehrnoori, 1980]. In addition, scientists are now pushing the limits of available computer resources (even on large vector computers) for three-dimensional problems such as contaminant modeling
follow a modal reduction technique based on the Arnoldi algorithm to solve a general transient transport equation. This method has the potential to render problems with large nodal discretizations into equivalent systems of much smaller size. Consequently, large savings in computer time are realized. Until now, prior application of modal reduction methods to transport problems has met with mixed results. Our focus in this paper is to demonstate the usefulnessof the algorithm for a series of hypothetical examples and to the Borden aquifer in Ontario, Canada.
in fractures.
In this paper we retain the finite element methodology but Copyright 1990 by the American Geophysical Union.
BACKGROUND
The transport processes considered in this paper are representedby the following differential equation, boundary, and initial conditions [Bear, 1972, pp. 613-617] V. (D' Vu) - v' Vu +f=
Ou/Ot
u(t = O) = Uo
in fl
in fl
u = uo
on
D. Vu. n = qo
on
(1)
where u is mass concentration, qt, is a specified mass flux, f is a source/sinkterm, D is a tensor of dispersioncoefficients, v is the transport velocity which depends on the spatial coordinates, n is the unit-outward normal, and t is time. 11 denotes
the
interior
of the
domain
under
consideration
Paper number 90WR00846.
whose boundary is 011.The componentsof D are commonly
0043-1397/90/90 WR-00846505.00
set to 2579
2580
WOODBURY ET AL.' ARNOLDI ALGORITHM APPLICATION TO CONVECTION-DISPERSION
Matrix Exponentials
viv•
=-vl0'+
+o*
D* is the coefficient of molecular diffusion in a porous medium, and the a terms are longitudinal and transverse dispersivity. A numerical solution of the above differential equation may be obtained by applying the finite element method, which results in the matrix differential equation Mfi + Ku = f where u is a vector
of n unknowns
(2)
at the nodes of the mesh
used to discretize fl, fi = Ou/Ot,M is the "capacity" matrix, and K is sometimes known as the "conductivity" matrix. The vector f contains the effects of sources, if present, and boundary conditions. M is symmetric and positive-definite. K is composedof a symmetricmatrix K•, which dependson D, and an unsymmetric matrix K2, which dependson the velocity v. There exist constraints on nodal spacing related to the magnitude of the velocity versus the dispersion coefficients. These constraints are to ensure the stability and nonoscillatory behavior of the computed solution [Carey and Sepehrnoori, 1980]. Beyond these constraints is a desire for a smooth and accurate solution, especially in regions likely to exhibit large gradients. Smoothnessand accuracy are determined by the mesh size and the order of the element. Refinement
of the mesh increases
the smoothness
and im-
An examination of (2) indicates that a direct analytic solution to the ordinary differential equation should be mathematically possible; however, in practice, this is not a recommended procedure. An example of this methodology is given by Loaiciga and Marino [1987]. Their solution is
u(t)= etcUo+ t e-rCEf(r) dr whereC = -M-1K and E = -M -1. The right-handside vector f is written as a function of time to include possible time-dependent sources, sinks, and boundary conditions. The vector of initial conditionsat the nodesis u0. Note that an analytic solutionof (2) involves matrix exponentialsof the
formetC, whereC is anunsymmetric matrix.Thedifficulties in evaluating matrix exponentials for either symmetric or unsymmetric matrices are described by Moler and VanLoan [1978] and Golub and VanLoan [1983, pp. 396-397]. Although considerableprogresshas been made in the development of methods for evaluating matrix exponentials, this approach becomes burdensome for large orders of C. The
operation countfora matrixexponential isapproximately n3 flops (floating point operations) per time evaluation. Numerical Laplace Transforms
Another examination of (2) indicates that a Laplace transform of the ordinary differential equation would eliminate the initial conditions and further transform the equation into another algebraicequation (with no time dependence)in the Laplace domain. For example,
proves the accuracy of the solution at the cost of introducing functions with high spatial frequencies. Our goal is to start with a sufficiently fine mesh and eliminate the highfrequency functions without compromisingthe smoothness (K + pM)6 = • (4) and accuracy. The usual approach to solve (2) is to construct a solution by finite difference approximation to the time derivative. For where u is the transformed vector of nodal concentrations, p is the Laplacevariableand• is the transformed right-hand instance, the Crank-Nicolson algorithm gives side vector. A solution in Laplace (p) space is formed and numerically transformed back to the time domain at desired M +-- K us+ • = M--- K us+ Atfs+•/2 (3) locations. A recent explanation and application of this 2 2 method is given by Sudicky [1989]. Sudicky refers to the where At is the time step between evaluation periods and s method as Laplace transform Galerkin (LTG). The method is the time level. Therefore t = sat. Provided the time step has the advantage that the solution is, in principle, continuremains constant during a simulation, the solution of (3) ous in time so that one can directly evaluate any number of involves an LU decompositionof the left-hand side matrix, nodal concentrations at any value of time without time followed by matrix-vector multiplications of the right-hand marching. However, a large overhead in computingmust be side and repeated back solves. Other approachesare possi- initially performed. As explained by Sudicky [1989], (4) must be repeatedly solved (say, up to 16 times) for different values ble and are discussed below. of the Laplace variable p. In addition, matrices K and M must be factorized in complex arithmetic for each evaluation. The factorization stepsmust be performed if boundary Iterative Solution Algorithms conditions or material properties are changed. For this In the case of extremely large problems, iterative solution reason, the method is not amemable to the solution of algorithmsoffer a practical alternative to LU decomposition nonlinear problems. and back solving [Schmid and Braess, 1988]. In such algorithms the major operations are repeated matrix-vector Rayleigh-Ritz Reduction multiplications. These are relatively easy to program, even One may reduce the size of the problem by means of when the entire system cannot reside in core [e.g., Carey and Jiang, 1986]. However, convergenceof these algorithms Rayleigh-Ritz methods. The physical basis of these methods
canonlybe Provedfor symmetri c positive definitematricesl Conjugategradientalgorithmshavealsobeendevelopedand appliedto problemsinvolvingunsymmetricmatrices[Jennings,1977;Axelsson,1989;Obeysekareet al., 1987].
is thatthe•ystemdefinedby (2) Can be accUrately described by a few "modes," muchlike the dynamicbehaviorof a linear structural system [Bathe, 1982, chapter 10]. The modescanbe definedin termsof the generalizedeigenvalue
WOODBURY
ET AL ' ARNOLDI
ALGORITHM
APPLICATION
TO CONVECTION-DISPERSION
2581
problem KZ = MZA, where Z is a matrix of eigenvectors Dunbar and Woodbury [ 1989] found that the Lanczos reduceach corresponding to a diagonal of the matrix of eigenval- tion offered significant advantages with respect to solution ues A. The reduction in size is achieved by using a subset time. Similar advantages can be expected for the transport m
