Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations

The Allen-Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time-discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen-Cahn equation obtained based on two first-order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.

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“…Using this lemma, the semi-discrete solution of ( 2) is shown to preserve MBP as stated below (see [19] Theorem 2 for the detailed proof).…”

Section: Preliminariesmentioning

confidence: 98%

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

Doan

Hoang

2023

Numerical Methods Partial

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“…Numerical experiments on three-dimensional coarsening dynamics demonstrated great computational efficiency and excellent parallel scalability of this approach on supercomputers. The overlapping localized ETD was analyzed in [22] and in [23] for the time-dependent diffusion and semi-linear parabolic equations respectively, in which the convergence of the iterative solutions to fully discrete localized ETD solutions and to the exact semi-discrete solution was rigorously proved. A non-overlapping localized ETD was proposed and analyzed for diffusion problems in [24], where the convergence and exact mass conservation were demonstrated.…”

Section: Introductionmentioning

confidence: 99%

Localized Exponential Time DifferencingMethod for Shallow Water Equations: Algorithms and Numerical Study

Meng¹

2021

CiCP

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In this paper, we investigate the performance of the exponential time differencing (ETD) method applied to the rotating shallow water equations. Comparing with explicit time stepping of the same order accuracy in time, the ETD algorithms could reduce the computational time in many cases by allowing the use of large time step sizes while still maintaining numerical stability. To accelerate the ETD simulations, we propose a localized approach that synthesizes the ETD method and overlapping domain decomposition. By dividing the original problem into many subdomain problems of smaller sizes and solving them locally, the proposed approach could speed up the calculation of matrix exponential vector products. Several standard test cases for shallow water equations of one or multiple layers are considered. The results show great potential of the localized ETD method for high-performance computing because each subdomain problem can be naturally solved in parallel at every time step.

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A parareal exponential integrator finite element method for semilinear parabolic equations

Huang,

Ju,

2024

Numerical Methods Partial

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In this article, we present a parareal exponential finite element method, with the help of variational formulation and parareal framework, for solving semilinear parabolic equations in rectangular domains. The model equation is first discretized in space using the finite element method with continuous piecewise multilinear rectangular basis functions, producing the semi‐discrete system. We then discretize the temporal direction using the explicit exponential Runge–Kutta approach accompanied by the parareal framework, resulting in the fully‐discrete numerical scheme. To further improve computational speed, we design a fast solver for our method based on tensor product spectral decomposition and fast Fourier transform. Under certain regularity assumption, we successfully derive optimal error estimates for the proposed parallel‐based method with respect to ‐norm. Extensive numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our method.

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Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations

Cited by 6 publications

References 32 publications

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

Localized Exponential Time DifferencingMethod for Shallow Water Equations: Algorithms and Numerical Study

A parareal exponential integrator finite element method for semilinear parabolic equations

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