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A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.

The study of travelling wave solutions of nonlinear partial differential equations (PDEs) is the major subject in many fields of physical and nonlinear sciences. Concepts like solitons, peakons, kinks, breathers, cusps and compactons have entered into various branches of natural sciences such as chemistry, biology, mathematics, communication and particularly in almost all branches of physics like the fluid dynamics, plasma physics, field theory, nonlinear optics and condensed matter physics. Among these nonlinear PDEs there exists an important class of the fifth order Korteweg-de Vries equations

where , and are real numbers. This class includes the well-known Lax [

In general, Equation (1) does not admit exact solutions, therefore one has to resort to numerical methods. Due to the fifth-order terms in these equations, it is very difficult to compute the solutions of these equations accurately and efficiently. Recently, Shen [

In this paper, we propose a discrete singular convolution method to solve fifth order Korteweg-de Vries equations. Discrete singular convolution (DSC) methods belong to the family of local spectral (LS) methods. They were proposed by Wei [

Recently, Pindza and Maré [

The discretization of the generalized Korteweg-de Vries equations in space with the DSC method yields a system of ordinary differential equations (ODE) that needs to be solved by time integration methods. We use the fourth order exponential time differencing Runge Kutta (ETDRK4) [

The layout of this paper is as follows. We describe the formulation of the DSC method in Section 2. In Section 3, we discuss the exponential time integration methods for solving the semi-discrete system resulting from the spatial discretization of the nonlinear PDEs. Numerical results illustrating the merits of the new scheme are given in Section 4 and we present our conclusions in Section 5.

Discrete singular convolution (DSC) methods are relatively new numerical techniques in the field of nonlinear equations. They are defined as follow. Consider a distribution, and an element of the space of test functions. A singular convolution can be defined by

where is a singular kernel. For many science and engineering problems, an appropriate choice of has to be done. For instance, in the field of interpolation of surfaces and curves the singular kernel of delta type is very important. For numerical solutions of partial differential equations, the kernel is essential, where the subscript n denotes the -th order derivative of the distribution with respect to parameter. While using the DSC method, numerical approximations of a function and its derivatives can be treated as convolutions with some kernels. According to the DSC method, the -th derivative of a function can be approximated as [

where is the grid spacing, is the set of discrete grid points which are centered around, and is the effective kernel, or computational bandwidth; and is usually smaller than the whole computational domain.

In the present paper, we focus our attention on the regularized Shannon kernel (RSK)

to provide discrete approximations to the singular convolution kernels of the delta type (3). The required derivatives of the DSC kernels can be easily obtained using ([

The error estimation of the regularized Shannon kernel (RSK) delivers very small truncation errors when it uses the above convolution algorithm ([

Theorem 2.1 (Qian [

Then

where and

Here is the number of grid points. The error given by (6) decays exponentially with respect to the increase of the DSC band width

The proof of the above theorem is beyond the scope of this paper. We refer the reader to [

Using (4) and (5), the entries of the first, second, third and fifth differentiation matrices, , and are given explicitly by

Note that the differentiation matrix in (5) is in general banded. This gives rise to great advantage in large scale computations. Extension to higher dimensions can be realized by tensor products.

The choice of, and was suggested by Qian and Wei [

where and is the frequency bound of the underlying function.

To illustrate the procedure of discretization of PDEs by the DSC method, we consider the computation of fifth order KdV equations given by

where, , and are real numbers and.

This equation was previously considered in [

The semi-discretized version, at the th row, of the equation in consideration is obtained by substituting the relations (3) and (5) into (12), yielding

where, , and are the typical elements of matrices, , and, respectively. Therefore, Equation (13) can be expressed in the following matrix form

where represents the linear part of the system and

represents the nonlinear part.

The main difficulty when dealing with systems of the type (14) is that the use of explicit time integrators is inefficient because the system typically suffers from instability due to the higher order derivative. This was emphasized by Pindza [

Exponential time differencing (ETD) schemes are known for a long time in computational electrodynamics; see [

The main idea of the ETD methods is to multiply both sides of a differential equation by some integrating factor, then we make a change of variable that allows us to solve the linear part exactly and, finally, we use a numerical method of our choice to solve the transformed nonlinear part.

In order to elaborate on this approach, let us consider the following semi-linear partial differential equation

where and are the linear and nonlinear operators, respectively. The semi-linear partial differential equation is discretized in space with the discrete singular convolution method. Therefore, we obtain a system of ordinary differential equations (ODEs)

The exponential time differencing (ETD) methods can be obtained by integrating Equation (16) exactly between the time steps and with respect to, to obtain

There exist various ETD methods for the evaluation of (17). The purpose of this work is not to give a complete classification of ETD methods. We focus specifically on the fourth order exponential time differencing RungeKutta (ETDRK4) given by

The main computational challenge in the implementation of exponential time differencing (ETD) methods is the need for fast and stable evaluations of exponential and related -functions

i.e., functions of the form. The computation of these functions depends significantly on the structure and the range of eigenvalues of the linear operator and the dimensionality of the semi-discretized PDE. Unfortunately, for DSC methods the linear part have eigenvalues approaching zero, which leads to complications in the computation of the coefficients. Saad [

Another way to compute the functions is to use the Taylor series representation. Therefore, for all complex numbers, we have

However, it is known that the computation of these functions in their explicit or Taylor series form suffers from computational inaccuracy for matrices whose eigenvalues are equal to or approaching zero. This is generally the case when the spatial discretization is based on spectral methods. In order to overcome the numerical difficulties encountered in the computation of (20) and (21), a different tactic for evaluating the function was proposed in [

where. If the contour encloses the spectrum of the non-diagonal matrix we have

If the size of the matrix is large, it is more advantageous to compute the product of the functions and vectors rather than to compute explicitly. We have

where and are the poles and the residues, respectively. The sum in (24) is evaluated by solving at most shifted linear systems. The poles and the residues are computed efficiently in standard precision by the Carathéodory-Fejér method [21,30].

In this section, we investigate the linear stability of the ETDRK4 method for the nonlinear autonomous system of ODEs,

linearized about a fixed point such that We obtain

where u is now the perturbation of and

is a diagonal or a block diagonal matrix containing the eigenvalues of. If, then the fixed point is stable for all.

The stability region is four-dimensional, if both and are complex. The two-dimensional stability region is obtained if both and are purely imaginary or purely real, or if is complex and is fixed and real.

In the paper, we follow the analysis employed in [

However, one can observe that the computation of, , and suffers from computational inaccuracy for values of equal to or approaching zero. Therefore, it is important to make use of their Taylor expansions

We commence our analysis by choosing real negative values of and looking for a region of stability in the complex plane where. Hence, the boundary of the stability region is determined by writing

The corresponding families of stability regions are plotted in the complex plane and displayed in

method for solving the fifth order KdV equation. To show the efficiency of the present method, we report the relative infinity and root mean square norm errors of the solution defined by

and

respectively, where is the number of interior points, and are the exact and computed values of the solution at point.

In this paper, we consider two case studies depending on the set of parameters of (25) that provide multi-soliton solutions. We evaluate the performance the DSC algorithms for different time increment, spatial discretization, the support size of DSC kernels and regularization parameter.

In our computation, the first set of parameters that we select are given by. In this case, the fifth order KdV Equation (11) is known as the SawadaKotera (SK) [

The SK (30) admits multi-soliton solutions [

where

respectively, with

In our computational work, we use the collocation points

The SK equation possesses infinite conservation laws [

related to the mass, momentum and energy. The quantities, and are applied to measure the conservation properties of the collocation scheme, calculated by

The second set of parameters are chosen as . This is well-known as the KaupKupershmidt (KK) [

Multi-soliton solutions can be generated by the following nonlinear transformation of the dependent variable,

For one soliton solution, the dependent variable function is given by

For two soliton solutions, the dependent variable function is

with

The KK equation possesses infinite conservation laws [

The quantities, and are applied to measure the conservation properties of the collocation scheme, calculated by

In next sections, we study the propagation and the interaction of single and two soliton solutions, respectively.

In our numerical experiments, we first model the motion of a single soliton of the SK (30) and KK (38) equations. For the SK equation, the initial condition is taken from the exact solutions (32) and (31) at initial profile. Whereas for the KK equation, the initial condition is taken from the exact solutions (40) and (39) at initial profile. The boundary conditions in both cases are chosen so that

In the first computation, we would like to investigate the convergence of the DSC method with respect to the number of grid points and the DSC bandwidth. The values of the parameters used in our numerical experiments are: and in both cases of the SK and KK equations. In each case, the soliton moves to the right across the space interval when the time interval is. The choice of the DSC bandwidth and the regularizer parameter is done according to the conditions (10). Hence if then. If then. If then.