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High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation. (English) Zbl 1156.65087

Summary: We introduce a high-order accurate method for solving a two dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing the spatial derivatives of the linear hyperbolic equation and a collocation method for the time component. The resulted method is unconditionally stable and solves the two-dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point the coefficients of which are determined by solving a system of linear equations. Numerical results show that the compact finite difference approximation of fourth order and the collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L15 Initial value problems for second-order hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:

[1] Mohanty, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Numer Methods Partial Differential Equations 17 pp 684– (2001) · Zbl 0990.65101
[2] Jezequel, A validated parallel across time and space solution of the heat transfer equation, Appl Numer Math 31 pp 65– (1999)
[3] Kouatchou, Finite differences and collocation methods for the solution of the two dimensional heat equation, Numer Methods Partial Differential Equations 17 pp 54– (2001) · Zbl 0967.65090
[4] Kouatchou, Parallel implementation of a high-order implicit collocation method for the heat equation, Math Comput Simul 54 pp 509– (2001) · Zbl 0987.68758
[5] Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int J Nonlinear Sci Numer Simul 7 pp 447– (2006) · Zbl 06942230
[6] Dehghan, A numerical method for solving the hyperbolic telegraph equation, Numer Methods Partial Differential Equations (2007)
[7] Gupta, A fourth-order Poisson solver, J Comp Phys 55 pp 166– (1984) · Zbl 0548.65075
[8] Dehghan, Multigrid solution of high order discreisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind, Appl Math Comput 180 pp 575– (2006) · Zbl 1102.65125
[9] Lapidus, Numerical solution of partial differential equations in science and engineering (1982) · Zbl 0584.65056
[10] Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math Comput Simulation 71 pp 16– (2006) · Zbl 1089.65085
[11] Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer Methods Partial Differ Equations 21 pp 24– (2005) · Zbl 1059.65072
[12] Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer Methods Partial Differ Equations 22 pp 220– (2006)
[13] Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons and Fractals 32 pp 661– (2007) · Zbl 1139.35352
[14] Dehghan, Numerical solution of the three-dimensional advection-diffusion equation, Appl Math Comput 150 pp 5– (2004) · Zbl 1038.65074
[15] Dehghan, Time-splitting procedures for the solution of the two-dimensional transport equation, Kybernetes 36 pp 791– (2007) · Zbl 1193.93013
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