Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Montanelli, Hadrien; Bootland, Niall

doi:10.1016/j.matcom.2020.06.008

Cited by 13 publications

(12 citation statements)

References 52 publications

(61 reference statements)

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“…There are a bunch of research related in AC and CH equations so far, some selected literatures are listed as follows. Montanelli and Bootland [15] proposed several exponential integration formula and compared their performance within stiff partial differential equations including AC and CH models. Such models are rewritten to sum of linear operator part with high-order terms and nonlinear operator part, and then Fourier-spectral method is applied in order to employ exponential integrator to this semilinear ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

Fourier-Spectral Method for the Phase-Field Equations

et al. 2020

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In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix.

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Section: Introductionmentioning

confidence: 99%

Fourier-Spectral Method for the Phase-Field Equations

et al. 2020

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“…We simulate (78) with periodic boundary conditions using the spin operator in the Chebfun package (www.chebfun.org, [104]). The solver uses exponential time differencing with fourthorder stiff time-stepping (ETDRK4, [105]); a survey and comparison of these methods is available from [106]. The same method is used to solve the other PDEs in this paper.…”

Section: Learning the Spectrum Of The Viscous Burgers' Equationmentioning

confidence: 99%

Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)

Baddoo,

Herrmann,

McKeon

et al. 2021

Preprint

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Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics, and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modeling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate. In contrast, sparse identification of nonlinear dynamics (SINDy) learns fully nonlinear models, disambiguating the linear and nonlinear effects, but is restricted to low-dimensional systems. In this work, we present a kernel method that learns interpretable data-driven models for high-dimensional, nonlinear systems. Our method performs kernel regression on a sparse dictionary of samples that appreciably contribute to the underlying dynamics. We show that this kernel method efficiently handles high-dimensional data and is flexible enough to incorporate partial knowledge of system physics. It is possible to accurately recover the linear model contribution with this approach, disambiguating the effects of the implicitly defined nonlinear terms, resulting in a DMD-like model that is robust to strongly nonlinear dynamics. We demonstrate our approach on data from a wide range of nonlinear ordinary and partial differential equations that arise in the physical sciences. This framework can be used for many practical engineering tasks such as model order reduction, diagnostics, prediction, control, and discovery of governing laws.

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“…Lawson methods exhibit a strong order reduction, in general. For particular problems, however, they show full order of convergence (see [2,4,5,13,18]). Most of the problems considered in these papers result from space discretizations of partial differential equations posed with periodic boundary conditions.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Unfortunately, Lawson methods exhibit a strong order reduction, in general. For particular problems, however, they show full order of convergence, see [2][3][4][5]18]. Lawson methods have also been used successfully in [1] for applications in optical fibres.…”

Section: Introductionmentioning

confidence: 99%

On the convergence of Lawson methods for semilinear stiff problems

2020

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Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schrödinger equation are included.

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Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Cited by 13 publications

References 52 publications

Fourier-Spectral Method for the Phase-Field Equations

Fourier-Spectral Method for the Phase-Field Equations

Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)

On the convergence of Lawson methods for semilinear stiff problems

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