Math @ Duke

Publications [#243719] of Anita T. Layton
Papers Published
 Bourlioux, A; Layton, AT; Minion, ML, Highorder multiimplicit spectral deferred correction methods for problems of reactive flow,
Journal of Computational Physics, vol. 189 no. 2
(2003),
pp. 651675 [doi]
(last updated on 2017/12/16)
Abstract: Models for reacting flow are typically based on advectiondiffusionreaction (ADR) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three subprocesses can be widely different, leading to disparate timestep requirements for robust and accurate timeintegration. in particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this timescale disparity. The proposed methods are highorder multiimplicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of ADR equations. Spectral deferred correction methods compute a highorder approximation to the solution of a differential equation by using a simple, loworder numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several subprocesses implicitly but independently, while avoiding the splitting errors present in traditional operatorsplitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semiimplicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the subprocess time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific timescales ordering, the generalization to any ordering combination is straightforward. © 2003 Elsevier Science B.V. All rights reserved.


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