## Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation.(English)Zbl 1183.65113

Summary: During the past few years, the idea of using meshless methods for numerical solution of partial differential equations has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. The meshless local Petrov-Galerkin (MLPG) method is one of the “truly meshless” methods since it does not require any background integration cells. The integrations are carried out locally over small sub-domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions.
In this paper the MLPG method for numerically solving the non-linear two-dimensional sine-Gordon (SG) equation is developed. A time-stepping method is employed to deal with the time derivative and a simple predictor-corrector scheme is performed to eliminate the non-linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of proposed method to deal with the unsteady non-linear problems in large domains.

### MSC:

 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations
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### References:

 [1] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Internat. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077 [2] Duarte, C.A.; Oden, J.T., H-p clouds-an hp meshless method, Numer. methods partial differential equations, 12, 673-705, (1996) · Zbl 0869.65069 [3] Liu, W.; Jun, S.; Zhang, Y., Reproducing kernel particle method, Internat. J. numer. methods fluids, 20, 1081-1106, (1995) · Zbl 0881.76072 [4] Lucy, L.B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 12, 1013-1024, (1977) [5] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. mech., 10, 307-318, (1992) · Zbl 0764.65068 [6] Melenk, J.M.; Babuska, I., The partition of unity finite element method: basic theory and applications, Comput. methods appl. mech. engrg., 139, 289-314, (1996) · Zbl 0881.65099 [7] Sukumar, N.; Moran, B.; Belytschko, T., The natural element method in solid mechanics, Internat. J. numer. methods engrg., 43, 839-887, (1998) · Zbl 0940.74078 [8] Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. comp., 37, 141-158, (1981) · Zbl 0469.41005 [9] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput. mech., 22, 2, 117-127, (1998) · Zbl 0932.76067 [10] Atluri, S.N.; Shen, S.P., The meshless local petrov – galerkin (MLPG) method: A simple and less-costly alternative to the finite element methods, Comput. model. eng. sci., 3, 1, 11-51, (2002) · Zbl 0996.65116 [11] Atluri, S.N., The meshless method (MLPG) for domain and BIE discretizations, (2004), Tech Science Press · Zbl 1105.65107 [12] Batra, R.C.; Porfiri, M.; Spinello, D., Treatment of material discontinuity in two meshless local petrov – galerkin (MLPG) formulations of axisymmetric transient heat conduction, Internat. J. numer. methods engrg., 61, 2461-2479, (2004) · Zbl 1075.80001 [13] Dehghan, M.; Mirzaei, D., The meshless local petrov – galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation, Eng. anal. bound. elem., 32, 747-756, (2008) · Zbl 1244.65139 [14] Dehghan, M.; Mirzaei, D., Meshless local petrov – galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. numer. math., 59, 1043-1058, (2009) · Zbl 1159.76034 [15] Sladek, J.; Sladek, V.; Krivacek, J.; Wen, P.H.; Zhang, Ch., Meshless local petrov – galerkin (MLPG) method for reissner – mindlin plates under dynamic load, Comput. methods appl. mech. engrg., 196, 2681-2691, (2007) · Zbl 1173.74482 [16] Sladek, J.; Sladek, V.; Hon, Y.C., Inverse heat conduction problems by meshless local petrov – galerkin method, Eng. anal. bound. elem., 30, 650-661, (2006) · Zbl 1195.80020 [17] Sladek, J.; Sladek, V.; Zhang, Ch.; Schanz, M., Meshless local petrov – galerkin method for continuously nonhomogeneous linear viscoelastic solids, Comput. mech., 37, 279-289, (2006) · Zbl 1103.74058 [18] Sladek, J.; Sladek, V.; Zhang, Ch.; Krivacek, J.; Wen, Ph., Analysis of orthotropic thick plates by meshless local petrov – galerkin (MLPG) method, Internat. J. numer. methods engrg., 67, 1830-1850, (2006) · Zbl 1127.74051 [19] Xiao, J.R.; McCarthy, M.A., Meshless analysis of the obstacle problem for beams by the MLPG method and subdomain variational formulations, Eur. J. mech. A solids, 22, 385-399, (2003) · Zbl 1032.74707 [20] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. simulation, 71, 16-30, (2006) · Zbl 1089.65085 [21] Pedersen, N.F.; Madsen, S., Search for the in-phase flux flow mode in stacked Josephson junctions, Physica C, 437, 262-266, (2006) [22] Darminto, H. Susanto; van Gils, S.A., Static sand dynamic properties of fluxons in a zig-zag $$0 - \pi$$ Josephson junction, Phys. lett. A, 361, 270-276, (2007) [23] Argyris, J.; Haase, M.; Heinrich, J.C., Finite element approximation to two-dimensional sine-Gordon solitons, Comput. methods appl. mech. engrg., 86, 1-26, (1991) · Zbl 0762.65073 [24] Bratsos, A.G., The solution of the two-dimensional sine-Gordon equation using the method of lines, J. comput. appl. math., 206, 251-277, (2007) · Zbl 1117.65126 [25] Djidjeli, K.; Price, W.G.; Twizell, E.H., Numerical solutions of a damped sine-Gordon equation in two space variables, J. engrg. math., 29, 347-369, (1995) · Zbl 0841.65083 [26] Christiansen, P.L.; Lomdahl, P.S., Numerical solutions of 2+1 dimensional sine-Gordon solitons, Physica D, 2, 3, 482-494, (1981) · Zbl 1194.65122 [27] Hirota, R., Exact three-soliton solution of the two-dimensional sine-Gordon equation, J. phys. soc. Japan, 35, 1566, (1973) [28] Christiansen, P.L.; Olsen, O.H., Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation, Phys. lett. A, 68, 2, 185-188, (1978) [29] Zagrodzinsky, J., Particular solutions of the sine-Gordon equation in 2+1 dimensions, Phys. lett. A, 72, 284-286, (1979) [30] Kaliappan, P.; Lakshmanan, M., Kadomtsev – petviashvili and two-dimensional sine-Gordon equations: reduction to painlevè transcendents, J. phys. A: math. gen., 12, L249-L252, (1979) · Zbl 0425.35080 [31] Leibbrandt, G., New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions, Phys. rev. lett., 41, 435-438, (1978) [32] Guo, B.Y.; Pascual, P.J.; Rodriguez, M.J.; Vázquez, L., Numerical solution of the sine-Gordon equation, Appl. math. comput., 18, 1-14, (1986) · Zbl 0622.65131 [33] Sheng, Q.; Khaliq, A.Q.M.; Voss, D.A., Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme, Math. comput. simulation, 68, 355-373, (2005) · Zbl 1073.65095 [34] Bratsos, A.G., An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions, Appl. num. anal. comp. math., 2, 2, 189-211, (2005) · Zbl 1075.65111 [35] Bratsos, A.G., A modified predictor – corrector scheme for the two-dimensional sine-Gordon equation, Numer. algorithms, 43, 295-308, (2006) · Zbl 1112.65077 [36] Bratsos, A.G., A third order numerical scheme for the two-dimensional sine-Gordon equation, Math. comput. simulation, 76, 271-282, (2007) · Zbl 1135.65358 [37] Bratsos, A.G., An improved numerical scheme for the sine-Gordon equation in 2+1 dimensions, Internat. J. numer. methods engrg., 75, 787-799, (2008) · Zbl 1195.78075 [38] Dehghan, M.; Mirzaei, D., The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Comput. methods appl. mech. engrg., 197, 476-486, (2008) · Zbl 1169.76401 [39] Mirzaei, D.; Dehghan, M., Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements, Eng. anal. bound. elem., 33, 12-24, (2009) · Zbl 1166.65390 [40] Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. comput. simulation, 79, 700-715, (2008) · Zbl 1155.65379 [41] Dehghan, M.; Mirzaei, D., The boundary integral equation approach for numerical solution of the one-dimensional sine-Gordon equation, Numer. methods partial differential equations, 24, 1405-1415, (2008) · Zbl 1153.65099 [42] Wazwaz, A.M., The tanh method and a variable separated ODE method for solving double sine-Gordon equation, Phys. lett. A, 350, 367-370, (2006) · Zbl 1195.65210 [43] Wazwaz, A.M., Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations, Chaos solitons fractals, 28, 127-135, (2006) · Zbl 1088.35544 [44] Helal, M.A., Soliton solution of some nonlinear partial differential equations and its application in fluid mechanics, Chaos solitons fractals, 13, 1917-1929, (2002) · Zbl 0997.35063 [45] Hu, D.A.; Long, S.Y.; Liu, K.Y.; Li, G.Y., A modified meshless local petrov – galerkin method to elasticity problems in computer modelling and simulation, Eng. anal. bound. elem., 30, 399-404, (2006) · Zbl 1187.74258 [46] Liu, K.Y.; Long, S.Y.; Li, G.Y., A simple and less-costly meshless local petrov – galerkin (MLPG) method for the dynamic fracture problem, Eng. anal. bound. elem., 30, 72-76, (2006) · Zbl 1195.74287 [47] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods appl. mech. engrg., 139, 3-47, (1996) · Zbl 0891.73075 [48] Bogolyubskiĭ, I.L.; Makhankov, V.G., Lifetime of pulsating solitons in certain classical models, JETP lett., 24, 1, 12-14, (1976) [49] Bogolyubskiĭ, I.L., Oscillating particle-like solutions of the nonlinear klein – gordon equation, JETP lett., 24, 10, 535-538, (1976) [50] Christiansen, P.L.; Grønbech-Jensen, N.; Lomdahl, P.S.; Malomed, B.A., Oscillations of eccentric pulsons, Phys. scr., 55, 131-134, (1997) [51] Malomed, B.A., Decay of shrinking solitons in multidimensional sine-Gordon equation, Physica D, 24, 155-171, (1987) · Zbl 0634.35077 [52] Malomed, B.A., Dynamic of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model, Physica D, 52, 157-170, (1991) · Zbl 0742.35060 [53] Maslov, E.M., Dynamics of rotationally symmetric solitons in near-SG field model with applications to large-area Josephson junctions and ferromagnets, Physica D, 15, 433-443, (1985) · Zbl 0583.35088 [54] Maslov, E.M., Rotationally symmetric SG oscillator with tunable frequency, Phys. lett. A, 131, 6, 364-367, (1988)
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