High order compact solution of the one-space-dimensional linear hyperbolic equation. (English) Zbl 1151.65071

Mohebbi, Akbar; Dehghan, Mehdi

Numer. Methods Partial Differ. Equations 24, No. 5, 1222-1235 (2008).

Summary: We introduce a high-order accurate method for solving one-space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivative of linear hyperbolic equation and collocation method for the time component. The main property of this method additional to its high-order accuracy due to the fourth order discretization of spatial derivative, is its unconditionally stability. In this technique the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations.
Numerical results show that the compact finite difference approximation of fourth order and collocation method produce a very efficient method for solving the one-space-dimensional linear hyperbolic equation. We compare the numerical results of this paper with numerical results of R. K. Mohanty [Appl. Math. Lett. 17, No. 1, 101–105 (2004; Zbl 1046.65076)].

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L15	Initial value problems for second-order hyperbolic equations

Keywords:

collocation; compact finite difference scheme; linear hyperbolic equation; telegraph equation; high accuracy

Citations:

Zbl 1046.65076

References:

[1]	Mohanty, On the use of high order difference methods for the system of one space second order non-linear hyperbolic equations with variable coefficients, J Comput Appl Math 72 pp 421– (1996)
[2]	Twizell, An explicit difference method for the wave equation with extended stability range, BIT 19 pp 378– (1979) · Zbl 0441.65066
[3]	Mohanty, An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation, Appl Math Lett 17 pp 101– (2004) · Zbl 1046.65076
[4]	Mohanty, An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl Math Comput 165 pp 229– (2005) · Zbl 1070.65076
[5]	Jezequel, A validated parallel across time and space solution of the heat transfer equation, Appl Numer Math 31 pp 65– (1999)
[6]	Kouatchou, Finite differences and collocation methods for the solution of the two dimensional heat equation, Numer Methods Partial Differ Equations 17 pp 54– (2001) · Zbl 0967.65090
[7]	Kouatchou, Parallel implementation of a high-order implicit collocation method for the heat equation, Math Comput Simulation 54 pp 509– (2001) · Zbl 0987.68758
[8]	Gupta, High-order difference schemes for two dimensional elliptic equations, Numer Methods Partial Differ Equations 1 pp 71– (1985)
[9]	Pozar, Microwave engineering (1990)
[10]	Dehghan, Multigrid solution of high order discretisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kinds, Appl Math Comput 180 pp 575– (2006) · Zbl 1102.65125
[11]	Lapidus, Numerical solution of partial differential equations in science and engineering (1982) · Zbl 0584.65056
[12]	Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int J Non-Linear Sci Numer Simul 7 pp 447– (2006) · Zbl 06942230
[13]	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
[14]	Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer Methods Partial Differ Equations 22 pp 220– (2006)
[15]	Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math Comput 71 pp 16– (2006) · Zbl 1089.65085
[16]	Shakeri, Numerical solution of the Klein-Gordon equation via He’s variational iteration method · Zbl 1179.81064
[17]	Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons and Fractals 32 pp 661– (2007) · Zbl 1139.35352
[18]	Dehghan, The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Computer Methods in Applied Mechanics and Engineering 197 pp 476– (2008) · Zbl 1169.76401