Evaluation of mixed Crank–Nicolson scheme and Tau method for the solution of Klein–Gordon equation
Introduction
The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. It represents the equation of motion of a quantum scalar or a pseudo-scalar field, which is a field whose quanta are spinless particles. Such a problem appears naturally in the study of some nonlinear dynamical problems of mathematical physics, such as radiation theory, general relativity, the scattering and stability of kinks, vortices and other coherent structures [11]. At first, this equation which describes relativistic electrons was described by two physicists Oskar Klein and Walter Gordon in 1926. The Klein–Gordon equation correctly describes the spinless pion, a composite particle where the Dirac equation describes the spinning electron. Schrodinger proposed the Klein–Gordon equation as a quantum wave equation in his search for an equation describing de Broglie waves in 1927. The fine structure of hydrogen atom is predicted incorrectly by this equation. A non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure is submitted by Schrodinger instead of his equation. A linear and nonhomogeneous Klein–Gordon equation has the form [6], [33] where □ is the d’Alembert operator, defined by . Also a nonlinear and nonhomogeneous Klein–Gordon equation is given by [10], [11], [13], [26], [36] This equation with is encountered in quantum field theory and a number of applications and is referred to as the nonlin ear Klein–Gordon equation so that some special cases are obtained by selecting the nonlinear force g(u) as
- •
Klein–Gordon equation with a power-law nonlinearity 1 [33], .
- •
Klein–Gordon equation with a power-law nonlinearity 2 [33], .
- •
Modified Liouville equation [5], [33], .
- •
Klein–Gordon equation with a exponential nonlinearity [33], .
- •
Sine-Gordon equation [16], [33], .
- •
Sine-Gordon equation [3], [33], .
With reference to the solution of this equation, we can see many published papers: The differential transform method is proposed for solving linear and nonlinear Klein–Gordon equation [36]. Authors of [26] studied the nonlinear Klein–Gordon equation with quadratic and cubic nonlinearities and proposed a numerical scheme to solve this equation using collocation and finite difference-collocation methods. In [26] b-spline scaling function is used for solving this equation. Numerical solution based on finite difference and collocation method of the nonlinear Klein–Gordon equation using radial basis functions is proposed in [13]. Four finite difference schemes for approximating the nonlinear Klein–Gordon equation have been presented in [25]. In [23] Li and Guo proposed a Legendre pseudospectral scheme for solving initial value problem of the nonlinear Klein–Gordon equation. Their numerical solution keeps the discrete conservation also the stability and convergence are investigated in this paper. Nonlinear Klein–Gordon equation on an unbounded domain is studied in [17]. In [17] Split local absorbing boundary (SLAB) conditions are obtained by the operator splitting method, then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. In [8], Adomians decomposition scheme is used as an alternate method for solving this equation. In [43], Wazwaz proposed the modified decomposition method for analytic treatment of nonlinear differential equations. The modified method accelerates the rapid convergence of the series solution and provides the exact solution by using few iterations only without any need to the so-called Adomian polynomials. In [43], nonlinear Klein–Gordon equation is solved by the modified decomposition method. In [24], Li and Quoc studied a formalism deriving systematically invariant, finite difference algorithms for nonlinear Klein–Gordon equation. The papers by [21], [22], [30], [38] may be consulted on the progress made during the last 20 years.
In this work, the nonlinear and nonhomogeneous Klein–Gordon equation is assumed as with initial conditions and boundary condition where u(x, t) is the unknown solution which must be found. g(u) satisfies the Lipschitz condition with respect to u where is Lipschitz constant.
The proposed equation will be reduced by Crank–Nicolson scheme to system of ordinary differential equations then the Tau method will be used to solve this system. Spectral Tau methods are similar to Galerkin methods in the way the differential equation is enforced. However, none of the test functions need to satisfy the boundary conditions. Hence, a supplementary set of equations is used to apply the boundary conditions [4], [7]. The Tau method have been extended in several directions such as: systems of linear differential equations [15], non-linear problems [31], [32]; partial differential equations [34]; and, in particular, to the numerical solution of non-linear systems of partial differential equations [19], to the numerical solution of Fredholm integro-differential equation and Volterra integral equation [20], [42].
In this work, interpolating scaling functions which are introduced by Alpert et al. [1], [2] are used. These functions are based on Legendre polynomials [37]. In addition to the simultaneous properties which proposed for multiwavelets, interpolating scaling functions have interpolating property. This feature helps to reduce the time and computational cost. Also operational matrix of derivative is derived in [12], [39] which helps to save computing time.
The organization of the paper is as follows, In Section 2, interpolating scaling functions are introduced and the operational matrix of derivative is obtained. In Section 3, we apply the proposed method to solve the nonlinear Klein–Gordon equation. The order of convergence is obtained in Section 4. The results of numerical experiments are presented in Section 5. Section 6 is dedicated to a brief conclusion.
Section snippets
Interpolation scaling functions
Due to some desirable properties, multi-wavelets have been found to be interesting rather than scaler wavelets [41]. The multi-wavelets used in this paper are Alpert’ multi-wavelets [1], [2]. In this work, the Alpert’ scaling functions known by the name of interpolating scaling functions are applied for solving Klein–Gordon equation. Assume Pr is the Legendre polynomial of order r and r is any fixed nonnegative integer number. Let τk denotes the roots of Pr for . Also suppose ωk is the
Description of the method
The general form of the method used in this article is based on the finite difference method and spectral method. We use θ-weighted scheme for variable t to reduce these equations to a system of ordinary differential equations [12], [26], then we solve this system by spectral methods (Tau methods) and find the solution of the given partial differential equation at the points (). Note that the partial derivative of u with respect to t can be approximated by a
Stability and convergence analysis
Here the energy method is used to analyze the stability of the time discretization by Crank–Nikolson scheme. For a nonnegative integer k, let denote the usual Sobolev space of real-valued functions defined on Ω. With this definition, the Sobolev spaces H2 admit a natural norm
where ⟨., .⟩ is the inner product of L2(Ω).
Lemma 4.1 For any functions f(x) , g(x), we have [9]
Theorem 4.1 Suppose g(u) satisfies Lipschitz condition with respect to u
Order of convergence
In the context of numerical methods, we think of as an error, and certainly much less than 1. Also UT and were supposed to be the exact and approximate Fourier coefficients of u at the special coordinates. Assume where the number C is called the asymptotic error constant and p is the order of convergence. For finding the order of convergence p and C, one can find the εm for . So for two consecutive iterations, we obtain Therefore p can
Test problems
In this section some computational results of numerical experiments will be illustrated with the method proposed in Section 3, to support our theoretical discussion. To show the efficiency of the present method for our problems in comparison with the exact solution, we report the L2 errors of the solution which is defined by where ui and are the exact and computed values of the solution u at time ti. Also to find the order of convergence, we use the l2-norm which is
Conclusion
In this article, a solution of the Klein–Gordon equation is studied. A numerical method is proposed to find its solution. The hybrid of the Crank–Nicolson scheme and the Tau method based on the interpolating scaling functions (ISFs) were used. We proved that the method used in this paper is unconditionally stable and convergent using the energy method. The new procedure was tested on several examples taken from the literature. Numerical simulations are reported to demonstrate the usefulness of
References (44)
- et al.
Adaptive solution of partial differential equations in multiwavelet bases
J. Comput. Phys.
(2002) - et al.
A decomposition method for solving the nonlinear Klein–Gordon equation
J. Comput. Phys
(1996) - et al.
Application of the dual reciprocity boundary integral equation technique to solve the nonlinear Klein–Gordon equation
Comput. Phys. Commun.
(2010) - et al.
Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions
J. Comput. Appl. Math.
(2009) - et al.
Split local absorbing conditions for one-dimensional nonlinear Klein–Gordon equation on unbounded domain
J. Comput. Phys.
(2008) - et al.
A Tau method based on non-uniform space-time elements for the numerical simulation of solitons
Computers Math. Appl.
(1991) - et al.
Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases
Appl. Math. Model.
(2003) - et al.
Numerical solution of Klein–Gordon and sine-Gordon equations by meshless method of lines
Eng. Anal. Bound. Elem.
(2013) - et al.
Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method
Appl. Math. Comput.
(2010) Large amplitude instability in finite difference approximations to the Klein–Gordon equation
Appl. Numer. Math.
(1999)