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Mathematics 2017, 5(4), 63; https://doi.org/10.3390/math5040063

Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations

Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA
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Received: 26 September 2017 / Revised: 7 November 2017 / Accepted: 8 November 2017 / Published: 16 November 2017
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Abstract

Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs. View Full-Text
Keywords: fourth-order partial differential equations (PDEs); implicit integration factor (IIF) methods; Krylov subspace approximation; high-order accuracy fourth-order partial differential equations (PDEs); implicit integration factor (IIF) methods; Krylov subspace approximation; high-order accuracy
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Machen, M.; Zhang, Y.-T. Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations. Mathematics 2017, 5, 63.

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