Application of new quintic polynomial B-spline approximation for numerical investigation of Kuramoto–Sivashinsky equation

A spline is a piecewise defined special function that is usually comprised of polynomials of a certain degree. These polynomials are supposed to generate a smooth curve by connecting at given data points. In this work, an application of fifth degree basis spline functions is presented for a numerical investigation of the Kuramoto–Sivashinsky equation. The finite forward difference formula is used for temporal integration, whereas the basis splines, together with a new approximation for fourth order spatial derivative, are brought into play for discretization in space direction. In order to corroborate the presented numerical algorithm, some test problems are considered and the computational results are compared with existing methods.


Introduction
The Kuramoto-Sivashinsky (KS) equation, a canonical nonlinear evolution equation, crops up in mathematical modeling of several physical phenomena indicating reactiondiffusion systems, unstable drift waves in plasmas, pattern formation on thin hydrodynamic films, flame front instability, long waves on the interface between two viscous fluids, fluid flow on a vertical plate and spatially uniform oscillating chemical reaction in some homogeneous medium [1,2]. The KS equation has chaotic behavior and exhibits a traveling wave like solution that moves without changing its shape in a finite spatial domain [3][4][5]. The generalized KS equation is given by subject to the following conditions: where y(x, t) gives the wave displacement at position x and time t, α, β, γ are constants and φ(x), φ i (t), ψ i (t) are known functions. The term y xx is responsible for instability at broad scales and the dissipative term y xxxx controls the damping effect at small scales. The nonlinear term yy x serves as an energy stabilizer by transmitting it between small and large scales [6]. The nonlinear evolution equations have attracted a considerable amount of research work in recent years [7][8][9][10][11][12][13][14][15]. Several numerical and analytical techniques have been proposed for solving these equations [16][17][18][19][20][21][22][23]. Khater and Temsah [24] employed the Chebyshev spectral collocation approach for an approximate solution of the generalized fourth order KS equation. Lai and Ma [25] proposed a lattice Boltzmann model for solving the nonlinear KS equation, A mesh free approach based on radial basis functions was used in [26] for an approximate solution of the generalized KS equation. Mittal and Arora [27] explored the numerical solution to KS equation by means of the Crank-Nicolson scheme and quintic B-spline (QnBS) functions. Porshokouhi and Ghanbari [28] implemented a variational iteration method for series solution of KS equation. The authors in [29] presented a numerical approach based on basis spline functions for an approximate solution of the KS equation. Rageh et al. [30] implemented a restrictive Taylor approximation method to find a numerical solution for the KS equation. Ersoy and Dag [31] proposed an exponential cubic B-spline method for numerical solution of KS equation. Mittal and Dahiya [6] proposed a differential quadrature method based on QnBS functions for solving the generalized KS equation. Gomes et al. [32] used linear feedback controls and techniques to stabilize the non-uniform unstable steady states of the generalized KS equation. The authors in [33] used polynomial scaling functions for solving the generalized KS equation. Akgul and Bonyah [34] proposed a reproducing kernel Hilbert space method for the solving generalized KS equation.
In this article, the numerical investigation of the nonlinear KS equation has been presented. The finite forward difference formulation and quintic polynomial basis spline functions are used to discretize the problem in time and spatial domains, respectively. The spatial order of convergence of a typical QnBS approximation scheme has been improved by involving a new approximation for the fourth order derivative. The stability of proposed algorithm has been studied by means of Von-Neumann stability analysis.
This study is organized as: In the first section, we discuss some basic ideas related to QnBS functions. The development of new approximation for y 4 (x) is explained in Sect. 3. The numerical method is discussed in Sect. 4. Section 5-6 consists of a stability and error analysis and finally the computational results are reported in Sect. 7.

Quintic polynomial B-spline functions
Let us partition the domain [a, b] into n intervals, [x i , x i+1 ], of equal length such that x i = a + (i × h), i = 0, 1, . . . , n, a = x 0 , b = x n and h = 1 n (ba). The rth polynomial B-spline of degree q, order q + 1, is defined as [35] B q,r (x) = L q,r B q- 1,r where q > 0, L q,r = (x-x r ) (x r+q -x r ) and Using (5)-(4) with q = 5, we get fifth degree basis spline functions [36]: where r = -2, -1, 0, . . . , n + 2. The QnBS approximation Y (x) for a sufficiently smooth function y(x) is given by where the σ r are to be calculated. Let Y i , m i , M i , T i and F i represent the QnBS approximations for y(x i ), y (x i ), y (2) (x i ), y (3) (x i ) and y (4) (x i ), respectively. Using (6) and (7), we have Moreover, from (8)- (12), we establish the following relations [37][38][39]: y (8) 6 30,240 y (9) We see that the truncation error in F i is O(h 2 ). Instead of using (12), the authors in [36,40] proposed a new O(h 3 ) accurate approximation for the fourth order derivative. For the sake of completeness, we reproduce those results in the following section.

Description of the numerical method
Applying a finite forward difference formula and θ weighted scheme in the time direction, the semi-discretized form of problem (1) is obtained as follows: where t is the mesh size in the time direction, 0 ≤ θ ≤ 1 and y j+1 is used to denote y(x, t j + t). The nonlinear term (yy x ) j+1 is treated as [33] (yy x ) j+1 = y j+1 y j x + y j y j+1 xy j y j x .
Consequently |ξ | ≤ 1, the proposed algorithm is proved to be stable.

Numerical results
To show the versatility of numerical algorithm, we have presented four numerical experiments. The accuracy and efficiency of the method is tested by the maximum, Euclidian and the global relative error (GRE) norms, which are calculated as [27,42] where Y i and y i are the approximate and exact solutions at the ith spatial knot, respectively. The numerical outcomes are compared with the Lattice Boltzmann model (LBM) [25], the Quintic B-spline collocation method (QnBSM) [27], B-spline functions (BSF) [29], the Exponential cubic B-spline collocation method (ExCBSM) [31], the QnBS differential quadrature method (QnBS-DQM) [6] and Polynomial scaling functions (PSF) [33].
The piecewise defined spline solution at t = 1 using proposed method for Example 3, when n = 100, t = 0.01 and λ = 0.1, is given by The exact solution is  Table 4. It is clear that our numerical algorithm provides a better approximation to the exact solution than BSF [29] and PSF [33]. The 2D graphs of numerical and true solutions at different time levels are shown in Fig. 5, and Fig. 6 depicts the 3D plots of the exact and numerical solutions in the temporal domain 0 ≤ t ≤ 10 using t = 0.01.   The piecewise defined spline solution at t = 1 using the proposed method for Example 4, when n = 100, t = 0.01, λ = 6, μ = 0.5 and ν = -10, is given by The exact solution is y(x, t) = 9 + λ -15 tanh(μ(xλtν) + tanh 2 (μ(xλtν) + tanh 3 (μ(xλtν) .

Conclusion
In this work, an application of a new quintic polynomial B-spline approximation approach has been presented for a numerical investigation of the Kuramoto-Sivashinsky equation. The numerical scheme employs typical fifth degree polynomial basis spline functions in association with a new approximation and a Crank-Nicolson scheme to discretize the problem in the space and time directions, respectively. The error and stability analysis of the proposed scheme is carried out. Four test problems are considered from the available literature and the simulation results are compared with LBM [25], QnBSM [27], BSF [29], ExCBSM [31], QnBS-DQM [6] and PSF [33]. It is concluded that the presented algorithm outperforms the other variants on the topic with superior accuracy and straightforward implementation.