Browsing by Author "Dawson, Clint N."

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    A Domain Decomposition Method for Parabolic Equations Based on Finite Elements
    (1990-10) Dawson, Clint N.; Du, Qiang
    A domain decomposition procedure for solving parabolic partial differential equations is presented. In this scheme, a Galerkin finite element discretization on a rectangular mesh is used. Boundary values at subdomain interfaces are calculated by an implicit/explicit procedure. This solution serves as boundary data for fully implicit subdomain problems, which can be solved simultaneously. Thus, the scheme is non-iterative, and requires non-overlapping subdomains. Numerical results are presented for problems in two space dimensions.
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    A Finite Difference Domain Decomposition Algorithm for Numerical Solution of the Heat Equation
    (1990-10) Dawson, Clint N.; Du, Qiang; Dupont, T.F.
    A domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented. In this procedure, interface values between subdomains are found by an explicit finite difference formula. Once these values are calculated, interior values are determined by backward differencing in time. A natural extension of this method allows for the use of different time steps in different subdomains. Maximum norm error estimates for these procedures are derived, which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters.
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    A Finite Element Domain Decomposition Method for Parabolic
    (1990-10) Dawson, Clint N.; Du, Qiang
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    A priori error estimates of finite element models of systems of shallow water equations
    (1998) Martinez, Monica Lucia; Wheeler, Mary F.; Dawson, Clint N.
    In recent years, there has been much interest in the numerical solution of shallow water equations. The numerical procedure used to solve the shallow water equations must resolve the physics of the problem without introducing spurious oscillations or excessive numerical diffusion. Staggered-grid finite difference methods have been used extensively in modeling surface flow without introducing spurious oscillations. Finite element methods, permitting a high degree of grid flexibility for complex geometries and facilitating grid refinement near land boundaries to resolve important processes, have become much more prevalent. However, early finite element simulations of shallow water systems were plagued by spurious oscillations and the various methods introduced to eliminate these oscillations through artificial diffusion were generally unsuccessful due to excessive damping of physical components of the solution. Here, we give a brief overview on some finite element models of the shallow water equations, with particular attention given to the wave and characteristic formulations. In the literature, standard analysis, based on Fourier decompositions of these methods, has always neglected contributions from the nonlinear terms. We derive ${\cal L}\sp{\infty} ((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for both the continuous-time and discrete-time Galerkin approximation to the nonlinear wave model, finding these to be optimal in ${\cal H}\sp1(\Omega).$ Finally, we derive ${\cal L}\sp{\infty}((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for our proposed Characteristic-Galerkin approximation to the nonlinear primitive model. We find these estimates to be optimal in ${\cal H}\sp1(\Omega)$ but with less restrictive time-step constraints when compared to the Galerkin estimates for the wave model.
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    Analysis of an Upwind-Mixed Finite Element Method for Nonlinear Contaminant Transport Equations
    (1993-11) Dawson, Clint N.
    In this paper, the numerical approximation of a nonlinear diffusion equation arising in contaminant transport is studied. The equation is characterized by advection, diffusion, and adsorption. Assuming the adsorption term is modeled by a Freundlich isotherm, it can be nonlinear in concentration and nondifferentiable as the concentration approaches zero. We consider the approximation of this equation by a method which upwinds the advection and incorporates diffusion using a mixed finite element method. Error estimates for a semi-discrete formulation are derived, and numerical results for a fully discrete formulation are given.
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    Analysis of Explicit/Implicit, Block Centered Finite Difference Domain Decomposition Procedures for Parabolic Problems
    (1991-11) Dawson, Clint N.; Dupont, Todd F.
    Domain decomposition procedures for solving parabolic equations are considered. The underlying discretization is block-centered finite differences. In this procedures, fluxes at subdomain interfaces are calculated from the solution at the previous time level. These fluxes serve as Neumann boundary data for implicit, block-centered discretizations in the subdomains. A priori error estimates are derived, and numerical results examining the stability, accuracy, and parallelism of the scheme are presented.
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    Effects of Lag and Maximum Growth in Contaminant Transport and Biodegradation Modeling
    (1992-04) Wood, Brian D.; Dawson, Clint N.
    The effects of time lag and maximum microbial growth on biodegradation in contaminant transport are discussed. A mathematical model is formulated that accounts for these effects, and a numerical case study is presented that demonstrates how lag influences biodegradation.
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    Explicit/Implicit, Conservative, Galerkin Domain Decomposition Procedures for Parabolic Problems
    (1990-10) Dawson, Clint N.; Dupont, Todd F.
    Several domain decomposition methods for approximating solutions of parabolic problems are given. These methods rely on implicit Galerkin procedures in the subdomains and explicit flux calculation on the inter-domain boundaries. The procedures are conservative both in the subdomains and across inter-domain boundaries. A priori error bounds and experimental results are presented.
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    Finite Element Approximations to the System of Shallow Water Equations, Part I: Continuous Time a Priori Error Estimates
    (1995-12) Chippada, S.; Dawson, Clint N.; Martinez, M.L.; Wheeler, Mary F.
    Various sophisticated finite element models for surface water flow exist in the literature. Gray, Kolar, Luettich, Lynch and Westerink have developed a hydrodynamic model based on the generalized wave continuity equation (GWCE) formulation, and have formulated a Galerkin finite element procedure based on combining the GWCE with the nonconservative momentum equations. Numerical experiments suggest that this method is robust, accurate and suppresses spurious oscillations which plague other models. We analyze a slightly modified Galerkin model which uses the conservative momentum equations (CME). For this GWCE-CME system of equations, we present an a priori error estimate based on an L² projection.
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    High Resolution Upwind-Mixed Finite Element Methods for Advection-Diffusion Equations with Variable Time-Stepping
    (1994-06) Dawson, Clint N.
    Numerical methods for advection-diffusion equations are discussed based on approximating advection using a high-resolution upwind finite difference method, and incorporating diffusion using a mixed finite element method. In this approach, advection is approximated explicitly and diffusion implicitly. We first describe the basic procedure where each advection time-step is followed by a diffusion step. Because the explicit nature of the advective scheme requires a CFL time-step constraint, the basic procedure may be expensive, especially if the CFL constraint is severe. Two alternative time-stepping approaches are presented for improving computational efficiency while preserving accuracy. In the first approach, several advective time-steps are computed before taking a diffusion step. In the second approach, the advective time steps are also allowed to vary spatially. Numerical results for these three procedures for a model problem arising in flow through porous media are given.
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    Large Time Solution of an Initial Value Problem for a Generalized Burgers Equation
    (1993-09) Grundy, R.E.; Sachdev,P.L.; Dawson, Clint N.
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    Mixed finite element methods for variably saturated subsurface flow
    (1996) San Soucie, Carol Ann; Dawson, Clint N.; Wheeler, Mary F.
    The flow of water through variably saturated subsurface media is commonly modeled by Richards' equation, a nonlinear and possibly degenerate partial differential equation. Due to the nonlinearities, this equation is difficult to solve analytically and the literature reveals dozens of papers devoted to finding numerical solutions. However, the literature also reveals a lack of two important research topics. First, no a priori error analysis exists for one of the discretization schemes most often used in discretizing Richards' equation, cell-centered finite differences. The expanded mixed finite element method reduces to cell-centered finite differences for the case of the lowest-order discrete space and certain quadrature rules. Expanded mixed methods are useful because this simplification occurs even for the case of a full coefficient tensor. There has been no analysis of expanded mixed methods applied to Richards' equation. Second, no results from parallel computer codes have been published. With parallel computer technology, larger and more computationally intensive problems can be solved. However, in order to get good performance from these machines, programs must be designed specifically to take advantage of the parallelism. We present an analysis of the mixed finite element applied to Richards' equation accounting for the two types of degeneracies that can arise. We also consider and analyze a two-level method for handling some of the nonlinearities in the equation. Lastly, we present results from a parallel Richards' equation solve code that uses the expanded mixed method for discretization.
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    Modeling Contaminant Transport and Biodegradation in a Layered System
    (1993-03) Wood, Brian D.; Dawson, Clint N.; Szecsody, Jim E.; Streile, Gary P.
    The transport and biodegradation of an organic compound (quinoline) were studied in a system of layered porous media to examine the processes that affect microbial growth near hydraulic layer interfaces. From a set of independent experiments, a mathematical model for the microbial kinetics was developed to describe the time rate of change of the concentrations of two organic compounds (quinoline and its first degradation product), one electron acceptor (oxygen), and microorganisms. This mathematical model was incorporated into a two-dimensional numerical model for flow and transport, so that simulations of the laboratory system could be conducted and results compared with the observed data. The model was formulated from the single-phase perspective (i.e, it did not include mass-transfer limitations between the aqueous and microbial phases). These comparisons suggest that, for some systems, a single-phase model can adequately describe the reactive processes that occur between aqueous components and microorganisms. Microbial lag was explicitly accounted for in the degradation kinetics. For the system described here, the inclusion of microbial lag was important for describing transient concentration pulses observed in the low-conductivity layer.
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    Noniterative Domain Decomposition for Second Order Hyperbolic Problems
    (1993-05) Dawson, Clint N.; Dupont, Todd F.
    The solutions of damped second order hyperbolic problems can have smooth components which decay slowly and rough components which decay quickly. If the behavior of the solution is of interest on the time scale of the slowly-decaying modes, then implicit time stepping methods may be more efficient than explicit methods. We formulate and analyze a Galerkin method for approximating the solutions of second order hyperbolic problems. This method involves domain decomposition in its formulation rather than as a means of solving the elliptic problems that result at each time step when a usual implicit method is used.
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    Simulation of Nonlinear Contaminant Transport in Groundwater by a Higher Order Godunov-Mixed Finite Element Method
    (1991-04) Dawson, Clint N.
    We consider the numerical simulation of contaminant transport in groundwater where the mathematical model includes a nonlinear adsorption term. The method we describe combines a higher order Godunov scheme for advection with a mixed finite element method for diffusion. The method is formulated in one space dimension, and numerical results for equilibrium and nonequilibrium adsorption are presented.
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    The Performance of an Explicit/Implicit, Domain Decomposition Procedure for Parabolic Equations on an Intel Hypercube
    (1991-05) Dawson, Clint N.
    A domain decomposition procedure for parabolic equations is described. In this procedure, the computational domain is divided into nonoverlapping subdomains. The equation is discretized by finite differences in time, and in space, a Galerkin finite element method is used on each subdomain. Subdomain solutions are related by an explicit flux calculation on the interfaces between subdomains. The interface fluxes are calculated in a stable and accurate manner, thus no iterations between the interface and subdomains are required. The method has been implemented on an Intel iPSC/860 Hypercube, and comparisons between domain decomposition solutions and a fully implicit Galerkin solution are presented for a set of test problems.
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    Time-Split Methods for Advective Flow Problems in Multidimensions Based on Combining Godunov-Type Procedures with a Mixed Finite Element Method
    (1991-09) Dawson, Clint N.
    Time-split methods for multidimensional advection-diffusion equations are considered. In these methods, advection is approximated by a Godunov-type procedure, and diffusion is approximated by a low order mixed finite element method. This approach is currently being used by a number of researchers to model fluid flow. A general methodology is outlined and analyzed, then two particular schemes for calculating advective fluxes are discussed. The first approach is an unsplit, higher-order Godunov method. In this approach, a method of characteristics is used to calculate the advective flux, and time steps larger than a CFL time step are considered. In an appendix, a modification to the first approach that is second order in time is analyzed.
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    Time-Splitting Methods for Advection-Diffusion-Reaction Equations Arising in Contaminant Transport
    (1992-03) Dawson, Clint N.; Wheeler, Mary F.
    Two time-splitting methods for advection-diffusion-reaction problems are discussed and analyzed. Numerical results for the first approach applied to bioremediation of contaminants in groundwater are presented.