An efficient scheme for numerical solution of Burgers’ equation using quintic Hermite interpolating polynomials

A numerical scheme combining the features of quintic Hermite interpolating polynomials and orthogonal collocation method has been presented to solve the well-known non-linear Burgers’ equation. The quintic Hermite collocation method (QHCM) solves the non-linear Burgers’ equation directly without converting it into linear form using Hopf–Cole transformation. Stability of the QHCM has been checked using Eucledian and Supremum norms. Numerical values obtained from QHCM are compared with the values obtained from other techniques such as orthogonal collocation method, orthogonal collocation on finite elements and pdepe solver. Numerical values have been plotted using plane and surface plots to demonstrate the results graphically.


Introduction
The majority of the problems arising in the field of physics, engineering, chemistry and biology, etc. are modelled using linear or non-linear partial differential equations. One such type of equation having numerous applications in physics and engineering is Burgers' equation. It is a well-known non-linear problem which gives an insight into the relation between convection and diffusion.
A variety of numerical methods have been developed to solve the Burgers' equation, such as finite difference scheme [12,14], finite element method [2], quadratic B-spline [13,16], cubic B-spline [5,9], automatic differentiation [6], and modified Adomain method [1]. In the present study, numerical solution of Burgers' equation has been shown by applying the quintic Hermite collocation method directly, without transforming the non-linear form into the linear form using Hopf-Cole transformation. The paper is divided into six sections. Section 1 gives the introduction of Burgers' equation, whereas Sect. 2 discusses about QHCM and collocation points. In Sect. 3, application and implementation of QHCM are discussed. In Sect. 4, stability analysis is discussed and Sect. 5 gives the discussion of all the results obtained and finally in Sect. 6, the crust of the present study is concluded.

Quintic Hermite collocation method(QHCM)
Quintic Hermite collocation is one of the Hermite collocation method [10,17], where Hermite interpolating polynomials are used as base functions. The trial function is approximated by Hermite interpolating polynomials of the order 2k + 1(k > 0). It is the generalization of the Lagrange interpolation with polynomials that not only interpolate function at each node but also its consecutive derivatives. In general for real numbers x 1 < x 2 < x 3 < · · · < x k and all integers m 1 , m 2 , m 3 , . . . , m k greater than zero, there exists a unique polynomial of degree m 1 + m 2 + m 3 + · · · + m k − 1. In the present work, quintic Hermite interpolating polynomials which are of order 5 are used to approximate the trial function.
Quintic Hermite interpolating polynomials can be expressed in the following form [8]: where P j ,P j ,P j can be expressed as: Orthogonal collocation is applied within each subdomain by introducing a new variable ξ = After rearranging the terms, P j ,P j ,P j can also be written in simplified form as:

Collocation points
Choice of collocation points is an important characteristic to be well thought out while considering the technique of orthogonal collocation. Collocation points are taken to be the zeros of orthogonal polynomials. Basically, the discretization end points are taken to be 0 and 1. In QHCM, four interior collocation points are taken within each element to discretize the problem. These interior collocation points are the zeros of orthogonal polynomials such as Jacobi polynomials [3]. The basic recurrence formula for Jacobi polynomials P In the present, the zeros of Legendre polynomials have been taken as collocation points as a special case of Jacobi polynomials for α = β = 0.

Application of QHCM
To apply QHCM on system of equations of non-linear Burgers' equation defined by (1)-(3), the approximating function is defined as: where c i s are continuous function of t. As defined earlier, to apply collocation, a new variable ξ is introduced within each of the subdomains such that the trial function takes the form: The boundary conditions are defined on x = 0 and x = 1. At jth collocation point, the system of equations (1)-(3) can be expressed as: Initially, After application of QHCM, the system of equations defined from (1) to (3) transforms into a set of ordinary differential equations (ODEs) with four ODEs within each subinterval [x i−1 , x i ]. As the approximating function consists of quintic Hermite interpolating polynomials which have the property to interpolate the firstand second-order derivatives at node points, due to which the additional condition of continuity waives off. It reduces the system of partial differential equations into a system of ODEs instead of the system of differential algebraic equations as in orthogonal collocation on finite elements (OCFE) [4,7]. After implementation of QHCM, the system of equations defined in Eqs. (7)-(9) can be written as: where D is the differential operator, u is the vector of collocation solutions u i s and M is the coefficient matrix at jth collocation point. The Matrix structure for QHCM is shown in Fig. 1. M is the square matrix of order 4ne × 4ne, Du and u are vectors of order 4ne. Bandwidth of each subinterval except first and last in matrix M is of order 4 × 6 and bandwidth of first and last subinterval is of order 4 × 5 due to boundary conditions. (1) with the initial condition u(x, 0) = sin π x; the problem becomes

Problem 3.1 First solve the Burgers' equation
Initially, Boundary conditions, Problem 3.2 In the second problem, the initial condition is taken to be u( Initially, Boundary condition, After the application of quintic Hermite collocation method to the above system of equations, 4ne number of equations appear, with ne as the number of elements. The resulting system of equation has been solved numerically using MATLAB with ode15s system solver.

Stability analysis
In the present work, stability of the numerical method has been checked by Eucledian norm and supremum norm. Let E = u − u γ , where E defines the pointwise rate of error, u being the exact solution and u γ is where E i is the pointwise error, w i s are the corresponding weight functions and h γ is the length of the γ th subdomain.
The order of convergence can be determined by the lemma given in [11] and is mentioned below: where C is the generic constant.
Therefore, the order of convergence of quintic Hermite interpolation is of the order h 6 . The stability analysis has been performed on the basis of maximum and Eucledian norm and is shown in Tables 1 and 2 for different values of ε. It has been observed that both the norms lie between 0 and 1.

Results and discussion
All the numerical findings obtained in this study have been adequately described in this section. Problem 3.1 has been solved numerically using different techniques such as QHCM, OCFE [4,7], pdepe solver, OCM   [15,18] for different values of ε and τ . In Tables 3, 4, 5, 6 and 7, the comparison between exact and the numerical solution with different techniques has been shown. It has been observed from these tables that the numerical values obtained from QHCM agree well with the exact ones and this fact authenticates that quintic Hermite collocation method is better than OCFE, pdepe solver and OCM. In Fig. 2, the graphical view of the numerical solution has been shown for different values of ε and τ in the form of 2D plots. In Fig. 3, surface plots have been shown for the graphical view of the solution for different values of ε and all τ . It has been observed that in all the figures values are symmetric and lie between 0 and 1. From these surface plots, the smoothness of the solution can also be observed easily.      Tables 8, 9, 10, 11 and 12 and it can be easily observed that numerical values agree well with the exact ones for QHCM as compared to OCFE, pdepe solver and OCM. In Fig. 4, numerical solution has been shown in the form of 2D plots for different values of ε and τ . In Fig. 5, surface plots have been plotted for different values of ε and all τ . From both the problems, it has been observed that the values obtained from QHCM agree well with the exact solution as compared to the OCFE, OCM and pdepe solver. Since for OCFE one needs a large number of elements, pdepe solver gives oscillations for certain values of parameter and because of stiff system of equations OCM does not give good results. Error analysis has been performed by calculating relative error (RE) given in the following formula: Tables 13 and 14 for Problems 3.1 and 3.2, respectively. It has been observed from these tables that maximum R.E. for QHCM is of order 10 −2 . This simply shows that the values obtained from QHCM agree quite well with the exact ones up to a certain degree of accuracy.

Conclusions
The Hermite collocation method developed here solves the non-linear Burgers' equation directly without transforming it into the linear form using Hopf-Cole transformation. The numerical study on two problems shows the accuracy of QHCM by comparing the values with other techniques as OCFE, pdepe solver and OCM. Results obtained from both the problems agree fairly well with the exact ones for QHCM as compared to other techniques, which authenticate the applicability of QHCM in solving non-linear stiff system of partial differential equations. This method is efficient due to its simplicity and easily programmable nature. For stability reasons, U 2 and U ∞ norms have been calculated.