Krylov implicit integration factor methods for semilinear fourth-order equations
type: Item
instance of: scholarly article
title: Krylov implicit integration factor methods for semilinear fourth-order equations
zbMATH Open document ID: 1395.65025
author: Michael Machen
author: Yong-Tao Zhang
published in: Mathematics
publication date: 2018-07-05T00:00:00Z
review text: Summary: Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [\textit{T. Jiang} and \textit{Y.-T. Zhang}, J. Comput. Phys. 253, 368--388 (2013; Zbl 1349.65305)], 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.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 35L76
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65N15
zbMATH DE Number: 6898692
zbMATH Keywords: fourth-order partial differential equations (PDEs)
zbMATH Keywords: implicit integration factor (IIF) methods
zbMATH Keywords: Krylov subspace approximation
zbMATH Keywords: high-order accuracy
describes a project that uses: RKC
MaRDI profile type: Publication
full work available at URL: math5040063
OpenAlex ID: W2768140852
cites work: Total variation diminishing Runge-Kutta schemes
cites work: Strong stability-preserving high-order time discretization methods
cites work: Total-Variation-Diminishing Time Discretizations
cites work: Efficient implementation of essentially nonoscillatory shock-capturing schemes
cites work: High-order multi-implicit spectral deferred correction methods for problems of reactive flow.
cites work: Spectral deferred correction methods for ordinary differential equations
cites work: Arbitrary order Krylov deferred correction methods for differential algebraic equations
cites work: Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics.
cites work: Semi-implicit spectral deferred correction methods for ordinary differential equations
cites work: Implicit-Explicit Methods for Time-Dependent Partial Differential Equations
cites work: Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes
cites work: Additive Runge-Kutta schemes for convection-diffusion-reaction equations
cites work: RKC time-stepping for advection-diffusion-reaction problems
cites work: Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows
cites work: Integral deferred correction methods constructed with high order Runge–Kutta integrators
cites work: A new class of time discretization schemes for the solution of nonlinear PDEs
cites work: Exponential time differencing for stiff systems
cites work: Fourth-Order Time-Stepping for Stiff PDEs
cites work: An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow
cites work: Efficient semi-implicit schemes for stiff systems
cites work: Compact integration factor methods in high spatial dimensions
cites work: Compact integration factor methods for complex domains and adaptive mesh refinement
cites work: Array-representation integration factor method for high-dimensional systems
cites work: Semi-implicit integration factor methods on sparse grids for high-dimensional systems
cites work: Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations
cites work: Krylov implicit integration factor methods for spatial discretization on high-dimensional unstructured meshes: application to discontinuous Galerkin methods
cites work: Computational complexity study on Krylov integration factor WENO method for high spatial dimension convection-diffusion problems
cites work: Krylov integration factor method on sparse grids for high spatial dimension convection-diffusion equations
cites work: Q4421310
cites work: Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems
cites work: Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations
cites work: Krylov single-step implicit integration factor WENO methods for advection-diffusion-reaction equations
cites work: Numerical methods for stiff reaction-diffusion systems
cites work: Efficient implementation of weighted ENO schemes
cites work: Efficient Solution of Parabolic Equations by Krylov Approximation Methods
cites work: Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
cites work: Q4337625
cites work: On Krylov Subspace Approximations to the Matrix Exponential Operator
cites work: Local discontinuous Galerkin methods for the Cahn-Hilliard type equations
cites work: Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations
cites work: The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems
DOI: 10.3390/MATH5040063
description: scientific article; zbMATH DE number 6898692