Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices

Abstract In this paper, we develop a numerical method for the N -dimensional sine-Gordon equation using differentiation matrices, in the theoretical frame of matrix differential equations. Our method avoids calculating exponential matrices, is very intuitive and easy to express, and can be implemented without toil in any number of spatial dimensions. Although there is currently a vast literature on the numerical treatment of the one-dimensional sine-Gordon equation, the references for the two-dimensional case are much sparser, and virtually nonexistent for higher dimensions. We apply it to a battery of two-dimensional problems taken from the literature, showing that it largely outperforms the previously existing algorithms; while for three-dimensional problems, the results seem very promising.

[1]  A. Scott,et al.  The soliton: A new concept in applied science , 1973 .

[2]  L. Trefethen,et al.  THE EIGENVALUES OF SECOND-ORDER SPECTRAL DIFFERENTIATION MATRICES* , 1988 .

[3]  P. S. Lomdahl,et al.  Numerical study of 2+1 dimensional sine-Gordon solitons , 1981 .

[4]  O. Klein,et al.  Quantentheorie und fünfdimensionale Relativitätstheorie , 1926 .

[5]  Fernando Vadillo,et al.  The solution of two-dimensional advection–diffusion equations via operational matrices , 2013 .

[6]  Fernando Vadillo,et al.  An integrating factor for nonlinear Dirac equations , 2010, Comput. Phys. Commun..

[7]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[8]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[9]  G. Lamb.  Elements of soliton theory , 1980 .

[10]  Elsayed M. E. Elbarbary,et al.  Higher order pseudospectral differentiation matrices , 2005 .

[11]  Alexandre T. Filippov.  The Versatile Soliton , 2000 .

[12]  Satish C. Reddy,et al.  A MATLAB differentiation matrix suite , 2000, TOMS.

[13]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[14]  O. Klein,et al.  Quantum Theory and Five-Dimensional Theory of Relativity. (In German and English) , 1926 .

[15]  W. Gordon,et al.  Der Comptoneffekt nach der Schrödingerschen Theorie , 1926 .

[16]  Mehdi Dehghan,et al.  Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM) , 2010, Comput. Phys. Commun..

[17]  David Gottlieb,et al.  The Spectrum of the Chebyshev Collocation Operator for the Heat Equation , 1983 .

[18]  Francisco de la Hoz,et al.  A matrix-based numerical method for the simulation of the two-dimensional sine-Gordon equation, , 2012 .

[19]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[20]  W. G. Price,et al.  Numerical solutions of a damped Sine-Gordon equation in two space variables , 1995 .

[21]  D. Zwillinger.  Handbook of differential equations , 1990 .

[22]  Richard Bellman,et al.  Introduction to matrix analysis (2nd ed.) , 1997 .

[23]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

[24]  William E. Schiesser,et al.  A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab , 2009 .

[25]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[26]  Anatoli V. Andreev.  Atomic Spectroscopy: Introduction to the Theory of Hyperfine Structure , 2005 .

[27]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[28]  P. Drazin,et al.  Solitons: An Introduction , 1989 .

[29]  Mehdi Dehghan,et al.  Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation , 2010, J. Comput. Appl. Math..

[30]  V. Fock,et al.  Zur Schrödingerschen Wellenmechanik , 1926 .

[31]  Mehdi Dehghan,et al.  A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions , 2008, Math. Comput. Simul..

[32]  A. G. Bratsos.  An improved numerical scheme for the sine‐Gordon equation in 2+1 dimensions , 2008 .

[33]  Mehdi Dehghan,et al.  Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements , 2009 .

[34]  C. Loan.  The ubiquitous Kronecker product , 2000 .

[35]  A. G. Bratsos.  The solution of the two-dimensional sine-Gordon equation using the method of lines , 2007 .

[36]  Y. Pavlov,et al.  Solutions of the three-dimensional sine-Gordon equation , 2009 .

[37]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[38]  E. H. Twizell.  Computational methods for partial differential equations , 1984 .