Efficient Parallel Algorithms for Parabolic Problems

Abstract

Abstract. Domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation. Implicit schemes are used in order to march with large time steps. Parallelization is realized by approximating interface values using explicit computation. Various techniques are examined, including a multistep second order explicit scheme and a one-step high-order scheme. We show that the resulting schemes are of second order global accuracy in space, and stable in the sense of Osher or in L∞. They are optimized with respect to the parallel efficiency.Key words. parabolic equations, finite difference, parallel efficiency, stability, approximation accuracy AMS subject classifications. 65N10, 65P05PII. S00361429003817101. Introduction. Domain decomposition is a powerful tool for devising parallel PDE methods. There is rich literature on domain decomposition methods for both elliptic and time-dependent problems [2,6]. We consider the linear parabolic problem in this paper. Explicit schemes are often naturally parallel and also easy to implement, but they usually require small time steps because of stability constraints. Implicit schemes are necessary for finding steady state solutions or computing slowly unsteady problems where one needs to march with large time steps; however, implicit schemes are not inherently parallel because at each time step essentially an elliptic type of problem needs to be solved.A conventional approach of parallelizing the implicit schemes is to apply the elliptic-type domain decomposition based preconditioning methods to the problem arising from the semidiscretization at each time step. It is noted [1] that the resulting problem is well conditioned when the time step is small; nevertheless, small step size is not always desirable in situations where implicit schemes become necessary to use.An alternative approach of parallelization is to take advantage of previous time steps since the problem considered here is time dependent; see [3,7,12,13,14] for related discussions. Kuznetsov [7] proposed a modified approximation scheme of mixed type, where the standard second order implicit scheme is used inside each subdomain, while the explicit Euler scheme is applied to obtain the interface values on the new time level. Once the interface values are available, the global problem is fully decoupled and can thus be computed in parallel. A similar hybrid scheme was proposed in [3], where instead of using the same spacing h as for the interior points where the implicit scheme is applied, a larger spacing H D is used at each interface point where the explicit scheme is applied. Due to stability and accuracy requirements, both methods do not lead to satisfactory parallel efficiency as shown

Unconditional stability of alternating difference schemes with variable time steplengthes for dispersive equation

“…Many powerful methods have been presented in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]20]. In 1983, Evans and Abdullah [1] observed that the alternate use of different schemes with truncation errors of opposite signs can implement the parallel computation and has high accuracy and unconditional stability.…”

Nonoverlapping domain decomposition characteristic finite differences for three‐dimensional convection‐diffusion equations

“…It is well known that domain decomposition [8][9][10] is a powerful tool to solve the large-scale computation. Domain decomposition methods are mostly classified as an overlapping or a nonoverlapping subdomain approach.…”

“…A standard second order explicit scheme related with a large step H is applied to compute artificial interface values, and corresponding implicit schemes are carried out with a small step h in each subdomain. It is noted [8,10] that the explicit nature gives rise to a constraint involving the time step and an interface discretization parameter (H > h), however, this constraint is much less severe than that of a fully explicit scheme. Convergence analysis in l ∞ -norm is presented by Dawson, Du, and Dupont, while the maximum principle they used is not suitable for domain decomposition procedures of complex problems due to the difficulty in theoretical analysis and the reduce in order of accuracy.…”

Corrected explicit-implicit domain decomposition algorithms for two-dimensional semilinear parabolic equations

“…Once the interface values are available, the whole problem is decoupled completely and parallelization is achieved. We note that the additional information (numerical interface condition) on the boundaries between subdomains is usually not part of the original mathematical model and the physical problem, thereby different ways to generate the numerical boundary condition lead to various explicit-implicit domain decomposition (EIDD) methods, see [12][13][14][15] for related discussions. These EIDD methods are globally non-iterative, algorithmically simple, and computationally and communicationally efficient for each time level; however, they always suffer from temporal step-size restrictions.…”