Journal ArticleDOI
A note on fourth-order time stepping for stiff PDE via spectral method
TLDR
In this article, the authors used the exponential time differencing Runge-Kutta 4 method (ETDRK4) to solve the diagonal example of a well known nonlinear partial differential equation (PDE) in the form of Burgers' equation.Abstract:
In this note it is illustrated that the Exponential Time Differencing (ETD) scheme needs the least steps to achieve a given accuracy, offers a speedy method in calculation time, and has exceptional stability properties in solving a stiff type problem. Nonetheless, the celebrated and well established method like RungeKutta is still being applied as the basis of many efficient codes. However, the stiff type problems seem cannot be solved efficiently via some of these methods. This note overcomes such stiff type problem via the exponential method. Furthermore, the exponential time differencing Runge-Kutta 4 method (ETDRK4) is used to solve the diagonal example of a well known nonlinear partial differential equation (PDE) in the form of Burgers’ equation. In addition, we use Fourier transformation for solving Burgers’ equation. Mathematics Subject Classification: 65M70, 65Z05read more
Citations
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Numerical Solution for Kawahara Equation by Using Spectral Methods
TL;DR: In this paper, the diagonal example of Kawahara equation via the ETD Runge-Kutta 4 technique is solved using short Matlab programs, and the results show remarkable stability characteristics upon resolving nonlinear wave equations.
Posted Content
Solving Navier-Stokes equations coupled with a heat transfer equation using Bagarello's approach and the Hankel transform
TL;DR: In this paper, the authors investigated the dynamics of an incompressible fluid in a bounded connected domain, described by Navier-Stokes equations coupled with a heat transfer equation, by using a method inspired from the non-commutative strategy developed by Bagarello, (see Int. Jour. of Theoretical Physics, 43, issue 12 (2004), p. 2371 - 2394).
Journal ArticleDOI
Numerical Solution of the Nonlinear Wave Equation via Fourth-Order Time Stepping
TL;DR: In this article, the authors solved the non-diagonal example of Fisher equation via the exponential time differencing Runge-Kutta 4 method (ETDRK4).
Journal ArticleDOI
An efficient numerical technique for the solution of nonlinear heat equation via spectral method
TL;DR: In this paper, the authors solved the diagonal example of nonlinear heat equation via the exponential time difference Runge-Kutta 4 methods (ETDRK4), and implemented the method by short Matlab programs.
References
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Journal ArticleDOI
Exponential Time Differencing for Stiff Systems
Stephen M. Cox,Paul C. Matthews +1 more
TL;DR: A class of numerical methods for stiff systems, based on the method of exponential time differencing, is developed, with schemes with second- and higher-order accuracy, and new Runge?Kutta versions of these schemes are introduced.
Journal ArticleDOI
Fourth-Order Time-Stepping for Stiff PDEs
TL;DR: A modification of the exponential time-differencing fourth-order Runge--Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators.
Journal ArticleDOI
The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems
James M. Hyman,Basil Nicolaenko +1 more
TL;DR: In this paper, the authors characterized the transition to chaos of the solutions to the Kuramoto-Sivashinsky equation through extensive numerical simulation, and showed that the attracting solution manifolds undergo a complex bifurcation sequence including multimodal fixed points, invariant tori, traveling wave trains, and homoclinic orbits.
Journal ArticleDOI
Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants
Journal ArticleDOI
Generalized integrating factor methods for stiff PDEs
TL;DR: This work proposes a generalization of the IF method, and in particular construct multistep-type methods with several orders of magnitude improved accuracy, and presents a new fourth order ETDRK method with improved accuracy.
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