An Exponential Time Differencing Method for the Kuramoto-Sivashinsky Equation

An Exponential Time Differencing Method for the Kuramoto-Sivashinsky Equation

Gentian Zavalani

Contact

Polytechnic University of Tirana

An Exponential Time Differencing Method for the Kuramoto-Sivashinsky Equation

Article Fingerprint

ReserarchID

9ZQOL

An Exponential Time Differencing Method for the Kuramoto-Sivashinsky Equation Banner

AI TAKEAWAY

Connecting with the Eternal Ground

Abstract

The spectral methods offer very high spatial resolution for a wide range of nonlinear wave equations, so, for the best computational efficiency, it should be desirable to use also high order methods in time but without very strict restrictions on the step size by reason of numerical stability. In this paper we study the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method; this scheme was derived by Cox and Matthews in [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comp. Phys. 176 (2002) 430-455] and was modified by Kassam and Trefethen in [A. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comp. 26 (2005Comp. 26 ( ) 1214Comp. 26 ( -1233]]. We compute its amplification factor and plot its stability region, which gives us an explanation of its good behavior for dissipative and dispersive problems. We apply this method to the Kuramoto-Sivashinsky Equation obtaining excellent results.

References

24 Cites in Article

Reference Format

G Beylkin,J Keiser,L Vozovoi (1998). A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs.
J Certaine The Solution of Ordinary Dierential Equations with Large Time Constants.
A Friedli (1978). Verallgemeinerte Runge-Kutta Verfahren zur Loesung steifer Differentialgleichungssysteme.
Syvert Norsett (1969). An A-stable modification of the Adams-Bashforth methods.
C Klein (2008). Fourth Order Time-Stepping for Low Dispersion Korteweg-de Vries and Nonlinear Schro dinger Equations.
A Kassam,L Trefethen (2005). Fourth-Order Time Stepping for Stiff.
Dieter Armbruster,John Guckenheimer,Philip Holmes (1989). Kuramoto–Sivashinsky Dynamics on the Center–Unstable Manifold.
Daniel Michelson (1986). Steady solutions of the Kuramoto-Sivashinsky equation.
B Nicolaenko,B Scheurer,R Temam (1985). Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors.
R Burden,J Faires (2001). Numerical Analysis.
M Hochbruck,A Ostermann (2005). Explicit Exponential Runge-Kutta Methods for Semi-linear Parabolic Problems.
S Cox,P Matthews (2002). Exponential Time Differencing for Stiff Systems.
A Kassam (2004). High Order Time stepping for Stiff Semi-Linear Partial Differential Equations.
John Miles (1981). An Envelope Soliton Problem.
John Boyd (2001). Chebyshev & Fourier Spectral Methods.
C Canuto,M Hussain,A Quarteroni,T Zang (1988). Spectral Methods in Fluid Dynamics.
B Fornberg (1998). A Practical Guide to Pseudospectral Methods.
D Gottlieb,S Orszag (1977). Numerical Analysis of Spectral Methods: Theory and Applications.
Lloyd Trefethen (2000). Spectral Methods in MATLAB.
Allan Taflove,Korada Umashankar (1995). Users Code for the Finite-Difference Time-Domain Method.
Gregory Beylkin,James Keiser,Lev Vozovoi (1998). A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs.
David Mott,Elaine Oran,Bram Van Leer (2000). A Quasi-Steady-State Solver for the Stiff Ordinary Differential Equations of Reaction Kinetics.
Q Du,W Zhu (2005). Analysis and applications of the exponential time differencing schemes and their contour integration modifications.
B Fornberg,T Driscoll (1999). A fast spectral algorithm for nonlinear wave equations with linear dispersion.

Download References

Funding

No external funding was declared for this work.

Conflict of Interest

The authors declare no conflict of interest.

Ethical Approval

No ethics committee approval was required for this article type.

Data Availability

Not applicable for this article.

How to Cite This Article

Gentian Zavalani. 2014. \u201cAn Exponential Time Differencing Method for the Kuramoto-Sivashinsky Equation\u201d. Global Journal of Research in Engineering - I: Numerical Methods GJRE-I Volume 14 (GJRE Volume 14 Issue I1): .

More Citation Formats

Select Citation Style:

Download Citation

Download Article

GJRE Volume 14 Issue I1
Pg. 55- 62

Explore Journals Explore Volume Read This Issue

Journal Specifications

Crossref Journal DOI 10.17406/gjre

Print ISSN 0975-5861

e-ISSN 2249-4596

Keywords

etd ETDRK4 integrating factor kuramoto-sivashinsky stiff systems

Submission ReceivedDecember 13, 2013
Peer Review Double Blind
Handling Editor
Accepted January 3, 2014
Published January 15, 2014

Version of record

v1.2

Issue date

June 30, 2014

Language

Experiance in AR

Explore published articles in an immersive Augmented Reality environment. Our platform converts research papers into interactive 3D books, allowing readers to view and interact with content using AR and VR compatible devices.

View in VR

Read in 3D

Your published article is automatically converted into a realistic 3D book. Flip through pages and read research papers in a more engaging and interactive format.

View in 3D

Article Matrices

Total Score: 101

Country: Albania

Subject: Global Journal of Research in Engineering - I: Numerical Methods

Authors: Gentian Zavalani (PhD/Dr. count: 0)

View Count (all-time): 208

Total Views (Real + Logic): 4762

Total Downloads (simulated): 2219

Publish Date: 2014 06, Mon

Monthly Totals (Real + Logic):

Month 1: 61 views
Month 2: 68 views
Month 3: 37 views
Month 4: 37 views
Month 5: 60 views
Month 6: 36 views
Month 7: 34 views
Month 8: 51 views
Month 9: 33 views
Month 10: 32 views
Month 11: 29 views
Month 12: 27 views
Month 13: 36 views
Month 14: 30 views
Month 15: 40 views
Month 16: 31 views
Month 17: 29 views
Month 18: 36 views
Month 19: 23 views
Month 20: 25 views
Month 21: 25 views
Month 22: 34 views
Month 23: 29 views
Month 24: 44 views
Month 25: 26 views
Month 26: 25 views
Month 27: 16 views
Month 28: 36 views
Month 29: 34 views
Month 30: 38 views
Month 31: 46 views
Month 32: 25 views
Month 33: 28 views
Month 34: 22 views
Month 35: 36 views
Month 36: 30 views
Month 37: 30 views
Month 38: 19 views
Month 39: 14 views
Month 40: 13 views
Month 41: 46 views
Month 42: 23 views
Month 43: 44 views
Month 44: 23 views
Month 45: 12 views
Month 46: 29 views
Month 47: 36 views
Month 48: 36 views
Month 49: 23 views
Month 50: 24 views
Month 51: 20 views
Month 52: 45 views
Month 53: 30 views
Month 54: 46 views
Month 55: 46 views
Month 56: 41 views
Month 57: 23 views
Month 58: 24 views
Month 59: 26 views
Month 60: 30 views
Month 61: 14 views
Month 62: 42 views
Month 63: 40 views
Month 64: 32 views
Month 65: 18 views
Month 66: 18 views
Month 67: 19 views
Month 68: 28 views
Month 69: 26 views
Month 70: 25 views
Month 71: 36 views
Month 72: 29 views
Month 73: 12 views
Month 74: 41 views
Month 75: 45 views
Month 76: 38 views
Month 77: 28 views
Month 78: 14 views
Month 79: 22 views
Month 80: 32 views
Month 81: 23 views
Month 82: 43 views
Month 83: 46 views
Month 84: 38 views
Month 85: 32 views
Month 86: 43 views
Month 87: 36 views
Month 88: 18 views
Month 89: 40 views
Month 90: 19 views
Month 91: 35 views
Month 92: 12 views
Month 93: 31 views
Month 94: 34 views
Month 95: 20 views
Month 96: 37 views
Month 97: 48 views
Month 98: 25 views
Month 99: 26 views
Month 100: 43 views
Month 101: 27 views
Month 102: 35 views
Month 103: 14 views
Month 104: 22 views
Month 105: 35 views
Month 106: 31 views
Month 107: 35 views
Month 108: 48 views
Month 109: 41 views
Month 110: 22 views
Month 111: 31 views
Month 112: 16 views
Month 113: 27 views
Month 114: 25 views
Month 115: 23 views
Month 116: 26 views
Month 117: 49 views
Month 118: 24 views
Month 119: 23 views
Month 120: 18 views
Month 121: 35 views
Month 122: 25 views
Month 123: 16 views
Month 124: 48 views
Month 125: 46 views
Month 126: 43 views
Month 127: 18 views
Month 128: 26 views
Month 129: 37 views
Month 130: 22 views
Month 131: 36 views
Month 132: 16 views
Month 133: 15 views
Month 134: 33 views
Month 135: 36 views
Month 136: 41 views
Month 137: 34 views
Month 138: 45 views
Month 139: 46 views
Month 140: 36 views
Month 141: 63 views
Month 142: 82 views
Month 143: 78 views

Total Views: 4762

Total Downloads: 2219

2026 Trends

Published Article

Our website is actively being updated, and changes may occur frequently. Please clear your browser cache if needed. For feedback or error reporting, please email [email protected]