A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods
Parand K. *, Nikarya M.
Department of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran

 

† Corresponding author. E-mail: k_parand@sbu.ac.ir mehran.nikarya@gmail.com

Abstract
Abstract

The Klein-Gordon equation arises in many scientific areas of quantum mechanics and quantum field theory. In this paper a novel method based on spectral method and Jacobian free Newton method composed by generalized minimum residual (JFNGMRes) method with adaptive preconditioner will be introduced to solve nonlinear Klein-Gordon equation. In this work the nonlinear Klein-Gordon equation has been converted to a nonlinear system of algebraic equations using collocation method based on Bessel functions without any linearization, discretization and getting help of any other methods. Finally, by using JFNGMRes, solution of the nonlinear algebraic system will be achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the Klein-Gordon equation and compare our results with other methods.

1 Introduction

The nonlinear Klein-Gordon equation arises in many scientific areas such as electromagnetic interactions, the relativistic hydrogen spectrum, coulomb scattering, nonlinear optics, solid state physics and quantum field theory etc.[13] The Klein-Gordon equation plays the role of one of the fundamental equations of quantum field theory. This equation describes relativistic electrons and is a quantized version of the relativistic energy momentum relation.[46] This equation was first considered as a quantum wave equation for an equation describing de Broglie waves,[1,4,7] also has a great importance in relativistic quantum mechanics, which is used to describe spinless particles. The nonlinear Klein-Gordon equation has the general form:[89]

with the initial conditions:

and with the Dirichlet or Neumann type boundary conditions, where τ, α, γ are known constants. Equation (1) is called Klein-Gordon with quadratic nonlinearity if k = 2, with cubic nonlinearity if k = 3. The numerical study of the nonlinear Klein-Gordon equation has been carried out for last half Century and still it is an active area of research to develop some better numerical schemes to approximate its solution. In the past decades many researchers have solved this problem.[5,1018] Recently, several numerical techniques have been developed for solving the Klein-Gordon equation (1), for example Luo et al. have solved this problem by using a fourth-order compact and conservative scheme, they discretized using the integral method with variational limit in space and the multidimensional extended Runge-Kutta method in time.[19] Verma et al. have proposed a numerical scheme based on forward finite difference, QLM process and DQ method,[9] Bhrawy and Soubhy by using Legendre Gauss-Lobatto collocation solve linear and nonlinear Klein-Gordon.[20] Aimi and Panizzi have solved 1D Klein-Gordon equation by using boundary element-finite element method coupling procedure.[7] Raza et al. have solved nonlinear Klein-Gordon equation using Sobolev gradients.[21] Dong et al.[22] by using a time-splitting Fourier pseudo-spectral discretization solved this problem. Guo and Wang have solved this problem by using collocation method based on Jacobi polynomials.[23] Bao and Dong have solved this problem by using finite difference method.[24] Kumar et al. have introduced numerical computation by using Homotopy analysis method to solve Klein-Gordon equation.[25] Biswas et al. have found traveling wave solutions of the nonlinear dispersive Klein-Gordon equation.[26] Hussain et al. have solved this problem by meshless method and method of lines.[27] Jang[28] has solved this problem by using semi-analytical method. Jiwari[29] has solved this problem by Lagrange interpolation and modified cubic B-spline differential quadrature methods. Shaoa and Wu have introduced a numerical solution of the nonlinear Klein-Gordon equation using the Chebyshev tau meshless method.[30] Pekmen and Tezer-Sezgin have solved this problem by using DQM.[8] Chang and Liu have introduced an implicit Lie-group iterative scheme to solve this problem.[31] Guo et al. have solved this problem by element-free kp-Ritz method.[32] Mohebbi et al. have introduced a method based on applying fourth order time-stepping schemes in combination with discrete Fourier transform to solve Klein-Gordon equation.[33]

Now in this paper, we intend to solve the Klein-Gordon equation using a novel method based on Bessel functions of the first kind, spectral collocation method and Jacobian free Newton-Krylov sub-space methods. Recently, Bessel functions have been used to solve nonlinear ODE, IDE, and fractional differential equation,[3436] now we want to use them to solve nonlinear PDE namely Klein-Gordon equation. The rest of this paper is organized as follows: the function approximation, Bessel functions and spectral methods are introduced in Sec. 2. In Sec. 3 the JFNGMRes with adaptive preconditioner is described to solve nonlinear systems of algebraic equations. Then to show the advantages, applicability and reliability of proposed method we solve some examples of Klein-Gordon equation and compare our results with others in Sec. 4. Finally, the paper concludes in Sec. 5.

2 Function Approximation and Spectral Methods Based on Bessel Functions

Let ω is a certain weight function, therefore:

where

In particular, in Hilbert space , (u, v)ω = ∫Λ u(x)v(x)ω(x)dx is inner product. If ω = 1 can be omitted, and let . Now for nonnegative integer m, we set

with following norm, and semi-norm

For any real r ≥ 0, we define the space by the space interpolation as in Adams.[37]

In this paper we use Bessel function of the first kind Jn(x) as the basis functions of L2(Λ):

where Γ(λ) is the gamma function which is defined as follows:

Series (2) is convergent for all −∞ < x < ∞. Also, Eq. (2) is a solution of the below Sturm-Liouville equation:[34,38]

It is clear that, Jn(x) are linear independent. Some recursive relations of derivation are as follows:[38]

Let J = [J0(x), J1(x), J2(x), …, Jn(x)]T therefore J′ = DJ, where D is derivative operational matrix and is obtained by using Eq. (4):

where the a0, a1, a2, …., an will be obtained by an interpolation technique.

Let N be a positive integer, we define space

this is clear that .

Now consider -orthogonal projection that, for any :

or equivalently,

Therefore we can write:

So, for any vHr(Λ) and r ≥ 0 we have:[39]

Several papers have discussed about convergence of spectral methods[3942] and spectral methods to solve nonlinear Klein-Gordon equation.[4344] In a same way we can write about convergence of proposed method to solve Klein-Gordon equation as follows.

2.1 Spectral Collocation Method

Spectral methods, in the context of numerical schemes for solving differential equations, generically belong to the family of weighted residual methods (WRMs).[46] WRMs represent a particular group of approximation techniques, in which the residuals (or errors) are minimized in a certain way and thereby leading to specific methods including Galerkin, Petrov-Galerkin, collocation, and tau formulations. Consider the approximation of the following problem via spectral method:

where is the differential or integral operation, is a lower-order linear and/or nonlinear operator involving only derivatives (if exist) and f(x, t) is a function of variables x and t, with enough initial and boundary conditions. The starting point of the spectral methods is to approximate the solution u(x, t) by a finite summation:

where ϕn’s are the basis functions that we have chosen Jn(x) as basis function and the expansion’s coefficients must be determined. Substituting u with uM,N in Eq. (7) leads to the residual function:

The notion of the WRM is to force the residual function to zero by requiring:

where {ψk} are test functions, and ω is positive weight function. The choice of test functions results to a kind of the spectral methods.[4748] A method for forcing the residual function (9) to zero, is the collocation algorithm.[3536,49] In this method, by choosing Lagrange basis polynomials as test function, such that ψj(x) = Lj(x) and using Gauss quadrature rule in Eq. (11) we can write:[46]

According to Lagrange polynomials definition ψ1i(xq) = δiq and ψ2j(tp) = δjp. By choosing xq and tp as collocation points and ωp = ωq = 1:

In this paper, since the PDE (7) is nonlinear, the obtained system of equations (13) is nonlinear, too. In next section, we will describe how to solve this nonlinear system of equations.

3 Newton-Krylov Algorithm

Solving a nonlinear differential equation by spectral method directly (without linearization or discritization) leads to solving a nonlinear system of algebraic equations F(x) = 0, where is a function F(x) = (f1(x), f2(x), f3(x), …., fn(x))T and is a vector. So speed and accuracy of solving this nonlinear system is very important. Many works have been done to improve solving the nonlinear systems.[5054] One of the best methods to solve a nonlinear system is classical Newton’s iterative method:

where F′(x) = J(x) is the n × n Jacobian matrix. Therefore:

In fact in each iteration, a linear system must be solved:

In spectral methods to increase the accuracy, the number of equations must be increased, so often size of system of equation is large. But for large and complicated nonlinear systems, calculation and reordering J(xn) and solving obtained linear system in each iteration could be mostly time consuming. Hence, there are some improvements in Eq. (16), where J(xn) is calculated and linear system will be solved. For example some mathematicians used a fixed Jacobian matrix in every iteration or used different linear solver with several preconditioners. One of the good ideas, is to use the finite difference technique to approximate Jacobian-vector product:

where ε is a very small value. Jacobian-vector product, can be useful to approximate Jacobian matrix and matrix-vector product. Also, for large dimensions, iterative methods such as GMRes or BiCGSTAB are preferred over direct solvers.[5455] In this paper, a Jacobian-free Newton GMRes (JFNGMRes) with an adaptively preconditioner have been used to solve large nonlinear system of equations. This preconditioner has been introduced by Saad[53] and used in Ref. [54]. Using this finite difference technique and updating adaptive preconditioning improve the computations in Newton’s method and GMRes.[5354] Many researchers have used Newton-Krylov methods and Jacobian free Newton-Krylov methods to solve several nonlinear problems.[5357]

3.1 Algorithm of Preconditioned JFNGMRes

In this section we describe the Jacobian free Newton’s method alongside generalized minimum residual with adaptive preconditioner

Begin

Set k = 0 (iteration counter of Newton method) and an initial guess x0.

Select a nonsingular Matrix M0 = ηI as the preconditioner, where and In × n = diag(1, …, 1)

Begin of Newton’s iterations: repeat until ||F(xk)||2 < err.

Use GMRes method with Jacobian free formula to solve linear system J(xk)δk = F(xk).

r0 = F(xk) − (F(xk + εxk) − F(xk))/(ε), β = ||r0||2, v1 = r0/β and

forj = 1 to Ndo

vj = Mk · vj.

.

.

.

Update the preconditioner matrix:

fori = 1 to jdo

hij = ⟨vj + 1, vi⟩.

vj + 1 = vj + 1hij · vi.

hj + 1, j = ||vj + 1||2.

if hj + 1,j = 0 then goto step 4.c else .

Now the Hessenberg matrix is constructed, square Hessenberg matrix is obtained by elimination of the last row of .

VN × N = [v1, v2, …, vN].

Now the less-square linear problem yk = min ||ρe1Hy||2 muse be solved, where e1 = [1, 0, 0, …, 0]T.

Using N Givens rotations G = g1, g2, …, gN, H becomes a upper trigonal matrix.

Now set R = GH and W = βGe1.

By using simple backward method the Ryk = W can be solved.

δk = Vyk

Set

xk + 1 = xkδk

Compute F(xk + 1) and ΔFk = F(xk + 1) − F(xk)

Update the preconditioner matrix Mk

k = k + 1 end repeat.

END

In Refs. [5354] inner and outer preconditioners and Mk and how to help it to Newton-Krylov method have been discussed.

4 Solving Some Examples of Nonlinear Kilen-Gordon Equation

To show the accuracy, availability, reliability and convergent rate of present method to solve Klein-Gordon equation, several examples of Klein-Gordon equation will be solved in this section. To obtain the solutions, we first transfer the solving the nonlinear Klein-Gordon equation to a nonlinear system of algebraic equations by using collocation spectral method based on Bessel function of the first kind without any discritization and linearization methods. Then we solve this nonlinear algebraic system by using JFNGMRes method and acquire the solution of this PDE. In solving procedure of all examples that follow in this paper, we use roots of shifted Legendre polynomial Pn(x) as collocation points, and satisfy the initial conditions by adding and multiplying some terms to the basis functions, satisfy the boundary conditions in the nonlinear system of equations and the initial guess of the iterative JFNGMRes method is vector [0, 0, …, 0]T. Also, some error definitions used in this article are as bellow:

Table 1

Comparison of obtained results of presented method with N = 17, M = 17 and 11 JFNGMRes iterations, and results of Refs. [5, 8, 14] for example 1.

.
Table 2

The convergence rate of presented method to solve example 1.

.
Fig. 1 The obtained graphs of solution for example 2 for c = 0.3 and several α, β. (a) α = 0.1, β = 1; (b) α = 0.2, β = 1; (c) α = 0.1, β = 10.
Table 3

Comparison of presented method and results of B-spline DQ method[29] for solving kink wave equation with α = 0.2, β = 1 and c = 0.3.

.
Table 4

Error and convergence rate of presented method for solving kink wave equation with different number points.

.
Fig. 2 Obtained graphs of single soliton solution. (a) α = β = − 1, c = 2; (b) α = 0.3, β = 1, c = 0.25.
Table 5

Obtained results and convergence rate of presented method with and 4 JFNG iteration for several α, β and c = 2 for single soliton problem at t = 1.

.
5 Conclusions

In this paper a new numerical algorithm was proposed to solve nonlinear Klein-Gordon equation. This method uses the spectral collocation method, the Bessel functions of the first kind as basis function and roots of the shifted Legendre polynomials as collocation point to convert nonlinear Klein-Gordon equation to a nonlinear system of algebraic equations. Then this nonlinear system is solved by using the Jacobian free Newton and GMRes methods with an adaptive preconditioner updated in each iteration. The obtained nonlinear system from spectral methods usually is large and ill-condition, therefore, iterative methods such as GMRes are preferred over direct solvers. Now we use an adaptive preconditioner to enhance the convergent rate of JFNGMRes. As indicated in the presented examples, the solutions of the nonlinear systems are obtained in 3, 4 and 11 Newton iterations, also in all examples the initial guess of JFNGMRes is simple vector [0, 0, …, 0]T, that show the speed and power of the proposed method. Also the shown RMS and L errors in the presented tables and comparison with others methods show efficiently, applicability and reliability of collocation method based on the Bessel functions of the first kind. Some advantages of the presented method include high convergence rate of collocation method based on Bessel functions of the first kind to solve Klein–Gordon equation, few iterations for Newton method, simple initial guess for JFNGMRes method and no need for discretization and linearization and saving memory and processing. In general in this paper there are some novelties: (i) Using Bessel function as basis function in spectral methods to solve nonlinear PDE. (ii) Using spectral methods without any time discritization and linearization method to solve Klein-Gordon equation. (iii) using Jacobian free Newton method with adaptive preconditioned GMRes in spectral methods to solve Klein-Gordon equation.

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