Explicit/implicit, conservative domain decomposition procedures for parabolic problems based on block-centered finite differences
Clint Dawson,Todd F. Dupont +1 more
TL;DR: Domain decomposition procedures for solving parabolic equations are considered, and a priori error estimates are presented, and numerical results examining the stability, accuracy, and parallelism of the scheme are presented.
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Abstract: Domain decomposition procedures for solving parabolic equations are considered. The underlying discretization consists of block-centered finite differences. In these 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 presented, and numerical results examining the stability, accuracy, and parallelism of the scheme are presented.
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Figures
FIG. 1. Initial condition for stability test 
TABLE 1 Convergence of solution: u(x, t) = ui(x, y, t) 
TABLE 2 Convergence of diffusive flux: u(x, t) = ui(x, y, t) 
FIG. 2. Plot of IJUII versus t 
TABLE 4 Convergence of diffusive flux: u(x, t) = u2(x, y, t) 
TABLE 3 Convergence of solution: u(x, t) = u2(x, y, t)
Citations
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References
A mixed finite element method for 2-nd order elliptic problems
P. A. Raviart,J. M. Thomas +1 more
- 01 Jan 1977
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On convergence of block-centered finite differences for elliptic-problems
Alan Weiser,Mary Wheeler Fanett +1 more
TL;DR: In this paper, the authors consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains and demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids.
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A Finite Difference Domain Decomposition Algorithm for Numerical Solution of the Heat Equation
TL;DR: In this article, a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented, where interface values between subdomains are found by an explicit finite difference formula, and interior values are determined by backward differencing in time.
Explicit-/implicit conservative Galerkin domain decomposition procedures for parabolic problems
Clint Dawson,Todd F. Dupont +1 more
TL;DR: Several domain decomposition methods for approximating solutions of parabolic problems using implicit Galerkin procedures in the subdomains and explicit flux calculation on the inter-domain boundaries are given.
2. Finite Element and Finite Difference Methods for Continuous Flows in Porous Media
Thomas F. Russell,Mary F. Wheeler +1 more
- 01 Jan 1983
TL;DR: Miscible displacement is an enhanced oil-recovery process that involves injecting a solvent into a petroleum reservoir to displace resident oil. The process is mathematically described by convection-dominated parabolic partial differential equations for each chemical component in the system. Numerical procedures developed for simulating miscible displacement are reviewed.