Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations

Abstract In this paper, a numerical solution of the coupled Burgers' equations based on septic B-spline collocation method is presented. The scheme is based on the Crank–Nicolson formulation for time integration and septic B-spline functions for space integration. The method has been showed unconditionally stable by using Von-Neumann technique. The efficiency of this method is demonstrated by applying two test problems. The obtained numerical results are found in a good agreement with the exact solution. This method is efficient, powerful, and economical. It can also applicable to other linear and nonlinear partial differential equations.


Introduction
The coupled Burger equations originally derived by Esipov to study the model of polydisperse sedimentation (Esipov, 1995). It's a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions and colloids, under the effect of gravity (Nee and Duan, 1998).

Septic B-spline collocation method
Consider a mesh a ¼ x 0 hx 1 hx 2 hÁ Á Á hx n ¼ b as a uniform partition of the solution domain a x b; with h ¼ x j Àx jÀ1 ; j ¼ 1; 2; ; ; ; ; N: The septic B-spline basis functions B j ðxÞat knots x j given as below: where fB À3 ; B À2 ; B À1 ; B 0 ; B 1 ; :::; B Nþ1 ; B Nþ2 ; B Nþ3 g forms a basis over the region ½a; b: Each septic Bspline covers eight elements so that an element is covered by eight septic B-splines.

Solution of coupled burger equations
The coupled Burger equations is given by with the boundary conditions: and initial conditions: To apply the proposed method, the time derivative was discretized by forward finite difference approximation and using Crank-Nicolson approach to equations (2) and (3), which obtained: where k ¼ Dt; is the time step. The nonlinear terms in equations (6) and (7) are linearized using the form given by Rubin and Graves ((Rubin and Graves, 1975). Then the nonlinear terms are approximated as the below: by approximating uðx; tÞ and vðx; tÞ by using septic B-spline functions B j ðxÞ and the time dependent parameters d j ðtÞ and r j ðtÞ; for Uðx; tÞ and Vðx; tÞ respectively, so the approximate solution written as: using approximate function (9) and septic B-spline functions (1), the approximate values at the knots of UðxÞ; VðxÞ and their derivatives up to second order are determined in terms of the time parameters d j ðtÞ and r j ðtÞ respectively, as: (1) by substituting the approximate solution for U; V and its derivatives from equations (10), equations (6) and (7) yields the following difference equations with the unknowns d j ðtÞ and r j ðtÞ : where The systems in the equations (11) and (12) consists of 2N þ 2equation in2N þ 14unknowns. To get a unique solution to the systems, 12 additional constraints are required. These are obtained from the boundary conditions (4). Application the boundary conditions enables us to eliminate the parameters d nþ1 Nþ3 ; r nþ1 Nþ3 from the system. Thus, we have a system of dimension (2 N þ 2) Â (2 N þ 2), which is the septa-diagonal system that can be solved by any algorithm.

Initial values
To solve the system, we apply the initial conditions to determine: The approximate solution must satisfy the following: (i) It must agree with the initial conditions at the knotsx i : (ii) The derivatives of the approximate initial condition agree with the exact initialconditions at both ends of the range.
The initial conditions and the derivatives at the boundaries are used as below: ðV 0 Þðx 0 ; 0Þ ¼ 7 h ðÀr À3 À56r À2 À245r À1 þ 245r 1 þ 56r 2 þ r 3 Þ ¼ g 0 ðx 0 Þ; ðV 00 Þðx 0 ; 0Þ ¼ 42 h 2 ðr À3 þ 24r À2 þ 15r À1 À80r 0 þ 15r 1 þ 24r 2 þ r 3 Þ ¼ g 00 ðx 0 Þ; which is a septa-diagonal system for unknown initial values d 0 j and r 0 j of orderð2N þ 2Þ; after eliminating the values of dandr: This system can be solved by any algorithm. Once the initial vectors of parameters have been calculated, the numerical solution of coupled Burger equations U and V can be determined from the time evaluation of the vectors d j and r j by using the recurrence relations:

Stability analysis of the method
The stability analysis based on the von Neumann concept in which the growth factor of a typical Fourier mode defined as: where A; Bare the harmonics amplitude, i is the imaginary unit. / ¼ kh; k is the mode number, h is the element size, and g is the amplification factor of the schemes. The non-linear terms in the scheme are linearized by assuming the nonlinear terms as a constants k 1 and k 2 respectively. Atx ¼ x j ; the equations (11) and (12) can be rewritten as: þ a 7 d nþ1 jþ3 Àa 8 r nþ1 jÀ3 Àa 9 r nþ1 jÀ2 Àa 10 r nþ1 jÀ1 þ a 10 r nþ1 jþ1 þ a 9 r nþ1 jþ2 þ A 8 r nþ1 jþ3 ¼ a 11 d n jÀ3 þ a 12 d n jÀ2 þ a 13 d n jÀ1 þ a 14 d n j þ a 15 d n jþ1 þ a 16 d n jþ2 þ a 17 d n jþ3 þ a 8 r n jÀ3 þ a 9 r n jÀ2 þ a 10 r n jÀ1 Àa 10 r n jþ1 Àa 9 r n jþ2 ÀA 8 r n jþ3 ; where a 12 ¼ 120 þ 42 24k 2h 2 þ 7 56kk 1 2h k 1 þ 7 56kk 2 2h k 2 ; þ b 10 d n jÀ1 Àb 10 d n jþ1 Àb 9 d n jþ2 Àb 8 d n jþ3 ; where Substituting (13) into the difference equation (14), yields  Similarly, substituting (13) into the difference equation (15), results: where and From equations (16) and (17),jgj 1 hence the scheme is unconditionally stable.

Numerical tests and results of coupled burgers' equations
The performance of the proposed method was tested by using two numerical examples, in this section L 2 and L 1 error norms obtained by the following formulas: Test problem (1): Numerical solution of coupled Burgers equations (2) and (3) is calculated for k 1 ¼ À2; k 2 ¼ k 3 ¼ 1which leads (2) and (3) as: with the following initial and boundary conditions: The exact solution is In the first computation,L 2 and L 1 error norms att ¼ 0:1; k ¼ 0:001was computed with various values ofN: The corresponding results are presented in Table  1. Second computation, L 2 and L 1 error norms at time level t ¼ 1; N ¼ 200 with decreasing values of Dt was calculated, the results are showed in Table 2. The results of both computations are the same for uðx; tÞand vðx; tÞ; because of the symmetric initial and boundary conditions. Furthermore, comparison of the numerical results of the problem (1) with the results obtained from Raslan et al. (2016) for N ¼ 50; k ¼ 0:01; k 1 ¼ À2; k 2 ¼ k 3 ¼ 1 with different time t has been studied. The results are presented in Table 3.

Conclusions
In this paper, a numerical scheme for the nonlinear coupled Burger's equations has been proposed using a collocation method based on septic B-spline functions. The proposed method is unconditionally stable. The method has been evaluated by two test problems. The accuracy of the method has been measured by computing L 2 and L 1 error norms. The obtained numerical results are quite satisfactory and comparable with the analytic solution and better than the obtained numerical results in (Raslan et al., 2016). Based on the stability and accuracy of the proposed method, the method can be extended to solve various linear and nonlinear partial differential equations.

Disclosure statement
No potential conflict of interest was reported by the authors.