A numerical solution of the Burgers' equation using cubic B-splines

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Abstract

In the present paper numerical solutions of the one-dimensional Burgers' equation are obtained by a method based on collocation of cubic B-splines over finite elements. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are found in good agreement with exact solutions. Time-space integration of the Burgers' equation yields a system of difference equation which is shown to be unconditionally stable.

Introduction

Consider the one-dimensional Burgers' equationUt+UUx=λUxx,a⩽x⩽b,t⩾0with the initial conditionU(x,0)=f(x)a⩽x⩽band the boundary conditionsU(a,t)=β1,U(b,t)=β2,where λ>0 is the coefficient of kinematic viscosity, and β1, β2 and f(x) will be chosen in a later section.

Because of its similarity to the Navier–Stokes equation, Burgers' equation often arises in the mathematical modelling used to solve problems in fluid dynamics involving turbulence. Burgers' equation was first introduced by Bateman [1] when he mentioned it as worthy of study and gave its steady solutions. It was later treated by Burger [2] as a mathematical model for turbulence and after whom such an equation is widely referred to as Burgers' equation. Since then the equation has found applications in field as diverse as number theory, gas dynamics, heat conduction, elasticity, etc. Hopf [3] and Cole [4] showed independently that this equation can be transformed to the linear diffusion equation and solved exactly for an arbitrary initial condition. The exact solutions of the one dimensional Burgers' equation have been surveyed by Benton and Platzman [7]. In many cases these solutions involve infinite series which may converge very slowly for small values of the viscosity coefficient λ which correspond to steep wave fronts in the propagation of the dynamic wave-forms [5]. Many studies have been done on the numerical solutions of Burgers' equation to deal with solutions for the small values of λ. Some of the earlier numerical studies are documented as follows: a finite element method has been given by Caldwell et al. [15] to solve Burgers' equation by altering the size of the element at each stage using information from three previous steps. Moreover Caldwell and Smith [17] have discussed the comparison of a number of numerical approaches to the equation. Nguyen and Reynen [18] also suggested a space-time finite element method based on a least-square weak formulation using piecewise linear shape functions. A kind of finite element method based on weighted residual formulation was given by Varoğlu and Liam Finn [14] and demonstrated the high accuracy and the stability. Spline and B-spline functions together with some numerical technique have been used in getting the numerical solution of Burgers' equation recently. Rubin and Graves have used the spline function technique and quasi-linearisation for the numerical solution of the Burgers' equation in one space variable [8]. A cubic spline collocation procedure was developed for the Burgers' equation in the papers [10], [19]. The implicit-finite difference schemes together with cubic splines interpolating space derivatives in the Burgers' equation has been proposed in the papers [11], [12], [13], [22]. The B-spline Galerkin method and B-spline collocation methods have been setup for the numerical solution of the differential equations [20], [21]. In addition to finite difference and finite element methods, some others methods exist in the literature.

In the present method, we have proposed a type of the cubic B-spline collocation procedure in which nonlinear term in the equation is linearized by using the form introduced by the Rubin and Graves [8]. For the numerical procedure, time derivative is discretized in the usual finite difference scheme. Solution and its principal derivatives over the subinterval are approximated by the combination of the cubic B-splines and unknown element parameters. Using the values of the cubic B-splines, nodal values and its derivatives at the knots are expressed in terms of element parameters. Placing nodal values and its derivatives in the Burgers' equation result in system consisting of N+1 equations for N+3 parameters. The resulting system can be solved with Thomas algorithm [6] after the boundary conditions are applied.

Section snippets

Collocation method

The region [a,b] is partitioned into uniformly sized finite elements of length h by the knots xj such that a=x0<x1<⋯<xN=b. Let φm(x) be cubic B-splines with knots at the points xm, m=0,…,N. The set of splines {φ−1,φ0,φ1,…,φN,φN+1} forms a basis for functions defined over [a,b]. Thus, an approximation UN(x,t) to the exact solution U(x,t) can be expressed in terms of the cubic B-splines as trial functions:UN(x,t)=∑N+1m=−1δm(t)φm(x),where δm are time dependent quantities to be determined from

The stability analysis

The nonlinear term UUx of the Burgers' equation is linearized by taking U as a constant g. Substituting the Fourier mode δmn=ξneiβmh, where β is the mode number and h is the element size, into linearized form of Eq. (12), we haveξ=a−ibc+id,wherea=2cosβh+4−(1−θ)12h2λΔt(1−cosβh),b=(1−θ)6hgΔtsinβh,c=2cosβh+4−θ12h2λΔt(1−cosβh),d=θ6hgΔtsinβh.

Taking the modulus of Eq. (14) gives |ξ|⩽1, we find that the difference scheme (12) is unconditionally stable for θ∈[12,1].

The test problems

We now obtain numerical solutions of Burgers' equation for three standard problems. To measure the accuracy of the numerical method between numerical and exact ones we compute the weighted 1-norm |e|1 defined by|e|1=1Ni=1N−1|U(xi,tj)−Ui,j||U(xi,tj)|

(a) The first problem has initial conditionU(x,0)=sin(πx),0⩽x⩽1and boundary valuesU(0,t)=U(1,t)=0,t⩾0.

The theoretical solution of this problem was expressed as an infinite series by Cole [4]U(x,t)=4πλ∑j=1jIj12πλsin(jπx)exp(−j2π2λt)I012πλ+2∑j=1Ij1

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