Numerical Techniques for Unsteady Nonlinear Burgers Equation Based on Backward Differentiation Formulas

Vijitha Mukundan; Ashish Awasthi

doi:10.1515/nleng-2017-0068

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Numerical Techniques for Unsteady Nonlinear Burgers Equation Based on Backward Differentiation Formulas

Vijitha Mukundan und Ashish Awasthi

Veröffentlicht/Copyright: 2. Juni 2018

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Abstract

We introduce new numerical techniques for solving nonlinear unsteady Burgers equation. The numerical technique involves discretization of all variables except the time variable which converts nonlinear PDE into nonlinear ODE system. Stability of the nonlinear system is verified using Lyapunov’s stability criteria. Implicit stiff solvers backward differentiation formula of order one, two and three are used to solve the nonlinear ODE system. Four test problems are included to show the applicability of introduced numerical techniques. Numerical solutions so obtained are compared with solutions of existing schemes in literature. The proposed numerical schemes are found to be simple, accurate, fast, practical and superior to some existing methods.

Keywords: Burgers equation; Method of lines; Backward Differentiation Formula; Stability

1 Introduction

Consider the nonlinear unsteady Burgers equation

ut+uux=νuxx,x∈[a,b]andt∈[0,T](1.1)

with initial condition u(x,0) = f(x), a ≤ x ≤ b, and boundary conditions u(a,t) = 0, u(b,t) = 0, 0 ≤ t ≤ T.

Here ν > 0, is the viscosity parameter and f(x) is given smooth function. Burgers proposed this equation as a mathematical model of turbulence. It is the simplified form of Navier-Stoke’s equation. It has got application in various fields like gas dynamics [34], traffic flow [18], shock theory [35], cosmology [38] etc. This equation has a time dependent term, convection term and diffusion term and can be used to test several numerical algorithms for Navier-Stoke’s equation. Hence, mathematicians have shown much interest in finding its numerical as well as analytical solution. In 1915, Bateman [4] introduced this equation and gave steady state solution for the problem. Later on, in 1948, J. M. Burgers, a dutch physicist proposed this equation as a mathematical model for turbulence [6,7] and due to his vast contribution the equation is named after him. Among the various numerical methods finite difference scheme draws more attention due to its simplicity, accuracy and wide applicability. Kadalbajoo and Awasthi [16] used Crank-Nicolson scheme effectively to the nonlinear Burgers equation using Cole-Hopf transformation. Among others numerical methods based on finite difference are given in [2, 3, 5, 9, 17, 20, 22, 23, 33, 41], while other methods include finite element method [32,36,37], Petro-Galerkin method [12], local-RBF meshless method [13], B-spline finite element method [14, 21, 24], and automatic differentiation method [1]. Recently, lattice Boltzmann method [11], multi-quadric quasi-interpolation [10] and meshless method [40] have been used to solve the nonlinear Burgers equation. A comparison of MOL to finite difference technique is presented in [19]. M.MRashidi et al. [30] employed homotopy perturbation method for solving generalized Burger and Burger–Fisher equations, while in [31] homotopy analysis method is used to solve Burger and regularized long wave equations. In 2015, V. Mukundan and A. Awasthi [27] proposed effcient numerical techniques to solve Burgers equation which is based on finite difference schemes. They have used Cole-Hopf transformation to reduce Burgers equation to diffusion equation and solved by backward differentiation formulas. On the other hand in [28] authors used fourth order three stage Runge-Kutta method for solving the system of ODEs.

In this paper, nonlinear time dependent Burgers equation is treated directly by Method of lines (MOL)technique without using any transformation. In Method of lines only the derivatives along X-direction are approximated by finite differences and the procedure is explained in Section 2. This technique reduces the nonlinear PDE to nonlinear system of ODE’s. In Section 3 stability of this nonlinear system is verified by Lyapunov’s indirect Method. Using Taylor series expansion the nonlinear system is linearized and then solved by Backward Differentiation Formulas which has been explained in Section 4. Four test problems have been solved, numerical outcomes have been discussed, two and three dimensional figures have been presented and comparison with other numerical schemes are given in Section 5. Conclusions are presented in Section 6.

2 Proposed Scheme

In this paper, we have proposed some schemes for efficiently dealing with the time dependent nonlinear Burgers equation. First we apply method of lines technique by discretizing the X-direction. For this we divide the spatial direction into N+1 equally spaced points with space interval h= (b – a)/N. Spatial derivatives are approximated using central difference scheme as given below.

∂u∂x(xi,t)=ui+1(t)−ui−1(t)2h,i=1,2,…N−1.

∂2u∂x2(xi,t)=ui+1(t)−2ui(t)+ui−1(t)h2,i=1,2,…N−1.

Applying the boundary conditions u₀(t) = 0, u_N(t) = 0 and substituting in Eq. (1.1), we get

dui(t)dt=νh2ui+1(t)−2ui(t)+ui−1(t)−ui(t)2hui+1(t)−ui−1(t)(2.2)

ui(0)=u0(xi),i=1,2,…N−1(2.3)

where, u_i(t) is approximation of u(x_i,t).

ODE system arising from the substituition of U(t) = [u₁(t),....u_N–1(t)]^⊤ in Eq. (2.2) is given by

dUdt=F(U,t),U(0)=U0,(2.4)

where, F is a nonlinear function of U with elements f_i which are defined as:

fi(u1,u2,...,uN−1,t)=ui+1(λ1−λ2ui)−ui(2λ1−λ2ui−1)+λ1ui−1,i=1,2,…N−1λ1=ν(h)2,λ2=1(2h)(2.5)

We will analyze the nonlinear system of ordinary differential equations (2.4) for existence, uniqueness of solution and its stability.

Theorem I

Let F(U,t) be a continuous function. Then the initial value problem (IVP)

dUdt=F(U,t),U(t0)=a

admits a solution U = f(t) in |(t – t₀)| < δ for some δ > 0.

Proof

Since the functions defined above

fi(u1,u2,...,uN−1,t)=ui+1(λ1−λ2ui)−ui(2λ1−λ2ui−1)+λ1ui−1,i=1,2,…N−1

are clearly continuous we can assure that solution exist for this IVP. □

Theorem II

Let C¹ denote the class of all differentiable functions whose first derivative is continuous.

If F(U,t) ∈ C¹ then there exist one and only one solution to the IVP.

Proof

The partial derivatives of the functions given in eq (2.5) are

∂fi∂ui=−2λ1−λ2ui+1+λ2ui−1,i=1,2,…N−1∂fi∂ui+1=λ1−λ2ui,i=1,2,…N−1∂fi∂ui−1=λ1+λ2ui,i=1,2,…N−1∂fi∂uj=0,i=1,2,…N−1,j≠i−1,i,i+1

All the partial derivatives of the function exist and are continuous everywhere in the domain, thus F(U,t) ∈ C¹.

Hence the IVP has a unique solution

Stability of the nonlinear system will be investigated in the next section.

3 Stability Analysis

One of the most important mathematical tool for analysis of stability of nonlinear system is Lyapunov’s stability theory. It allows to test the stability of a nonlinear autonomous system by finding the eigenvalues of the Jacobian matrix at the equilibrium point.

Consider the nonlinear system Eq. (2.4). Since the system is autonomous we have

dUdt=F(U),U(0)=U0(3.6)

where F is a nonlinear function of U with elements f_i defined as:

fi(u1,u2,...,uN−1,t)=ui+1(λ1−λ2ui)−ui(2λ1−λ2ui−1)+λ1ui−1,i=1,2,…N−1

Expanding F as a Taylor series about the equilibrium point U^* = 0, we have

F(U)≅F(U∗)+F′(U∗)(U−U∗)≅F′(U∗)U

We will investigate the stability of the system Eq. (3.6) using Lyapunov’s Indirect Method.

3.1 Lyapunov’s Indirect Method

Let x = 0 be an equilibrium point of ẋ = f(x), where f : D ⟶ Rⁿ is a continuously differentiable and D is a neighborhood of the origin. Let

A=∂f∂x∣x=0

Then

The origin is asymptotically stable if Re(λ_i) > 0 for all eigenvalues λ_i of A
The origin is unstable if Re(λ_i) > 0 for some eigenvalues λ_i of A

For the nonlinear system Eq. (3.6) the Jacobian matrix is given by

JF(n)=(∂f1∂u1)(n)(∂f1∂u2)(n)…(∂f1∂uN−1)(n)⋮(∂fN−1∂u1)(n)(∂fN−1∂u2)(n)…(∂fN−1∂uN−1)(n)

Let matrix A denote Jacobian matrix evaluated at the equilibrium point. Then A is a tridiagonal matrix given by

A=−2λ1λ1λ1−2λ1λ1⋱⋱⋱λ1−2λ1λ1λ1−2λ1

Eigenvalues of this matrix are of the form

−2λ1+2λ1cossπn+1,s=1,2,..,n.

Since, –1 < cos (sπn+1)

< 1 ⇒ 2λ₁ cos (sπn+1) < 2λ₁.

We have – 2λ₁ + 2λ₁ cos

(sπn+1) < 0,

i.e all eigenvalues of the Jacobian matrix are in the open left half complex plane. Hence origin is asymptotically stable for the nonlinear system.

4 Numerical Integration

Divide the time interval into M + 1 equally spaced points with time step Δt = T/M. For time integration we use backward differentiation formulas.

4.1 Backward Differentiation Formula of order one (BDF-1)

Un+1=Un+(Δt)F(Un+1,tn+1),n=0,1,...M−1(4.7)

U⁰ is the initial condition and U(n)=[u1(n),....uN−1(n)]T..

Since the system (4.7) is nonlinear, we can linearize it by using Taylor series expansion about U⁽ⁿ⁾

F(U(n+1))=F(U(n))+JF(n)(U(n+1)−U(n))+O((Δt)2),(4.8)

where JF(n)

is the Jacobian matrix at the n^th time level.

Substituting Eq. (4.8) in Eq. (4.7) we get,

U(n+1)=U(n)+Δt[F(U(n))+JF(n)(U(n+1)−U(n))]U(n+1)=U(n)+(I−ΔtJF(n))−1Δt[F(U(n))](4.9)

4.2 Backward Differentiation Formula of order two (BDF-2)

U(n+1)=43Un−13Un−1+23(Δt)F(Un+1,tn+1),n=1,2,...M−1(4.10)

U¹ is attained from BDF-1. For linearisation we use Taylor series expansion

F(U(n+1))=F(U(n))+JF(n)(U(n+1)−U(n))+O((Δt)2),(4.11)

where JF(n)

is the Jacobian matrix at the n^th time level.

Substituting Eq. (4.11) in Eq. (4.10) we get,

U(n+1)=43U(n)−13U(n−1)+2Δt[F(U(n))+JF(n)(U(n+1)−U(n))]3(I−2Δt3JF(n))U(n+1)=(43I−2Δt3JF(n))U(n)+2ΔtF(U(n))3−13U(n−1)U(n+1)=(I−2Δt3JF(n))−1(43I−2Δt3JF(n))U(n)+(I−2Δt3JF(n))−12ΔtF(U(n))3−(I−2Δt3JF(n))−113U(n−1)(4.12)

Hence the above scheme is linearized. Now, we need only to solve linear algebraic equations Eq. (4.12) which take less computational time.

4.3 Backward Differentiation Formula of order three (BDF-3)

U(n+1)=1811Un−911Un−1+211Un−2+611(Δt)F(Un+1,tn+1),n=2,...M−1(4.13)

U¹ and U² are attained from BDF-1. For linearisation we use Taylor series expansion

F(U(n+1))=F(U(n))+JF(n)(U(n+1)−U(n))+O((Δt)2),(4.14)

where JF(n)

is the Jacobian matrix at the n^th time level.

Substituting Eq. (4.14) in Eq. (4.13) we get,

U(n+1)=1811U(n)−911U(n−1)+211U(n−2)+6Δt[F(U(n))+JF(n)(U(n+1)−U(n))]11

(I−6Δt11JF(n))U(n+1)=(1811I−6Δt11JF(n))U(n)+6ΔtF(U(n))11−911U(n−1)+211U(n−2)

U(n+1)=(I−6Δt11JF(n))−1(1811I−6Δt11JF(n))U(n)

+(I−6Δt11JF(n))−16ΔtF(U(n))11−(I−6Δt11JF(n))−1911U(n−1)+(I−6Δt11JF(n))−1211U(n−2)

where, JF(n)

is the Jacobian matrix at the n^th time level. Hence the above scheme is linearized. Again, we need only to solve linear algebraic equations which take less computation time.

5 Numerical results and discussion

Several test experiments were conducted to check the efficiency and adaptability of the proposed numerical schemes. A comparison is made between the computed solutions and the test problems whose exact solutions are known for different values of viscosity ν and at different values of final time. The numerical solutions are also compared with results given in the literature. All numerical calculations were performed using codes produced in MATLAB 8.0.

Example-1

Consider the Burgers Equation [8]

ut+uux=νuxx,x∈[0,1]andt∈[0,T]

with initial condition u(x, 0) = sin (πx), 0 ≤ x ≤ 1, and the boundary conditions u(0,t) = u(1,t)= 0, 0 ≤ t ≤ T.

The exact solution of the problem is

u(x,t)=2πν∑n=1∞Cnexp−n2π2νtnsin⁡(nπx)C0+∑n=1∞Cnexp−n2π2νtcos⁡(nπx),

C0=∫01exp⁡{−12πν[1−cos⁡(πx)]}dx,Cn=2∫01exp⁡{−12πν[1−cos⁡(πx)]}cos⁡(nπx)dx.

The Burgers equation has been solved directly by first applying method of lines semi discretization technique followed by implicit solvers backward differentiation formulas of order one, two and three. In Table 1, numerical solutions obtained by proposed method followed by backward differentiation formulas are compared with the exact solutions for viscosity ν= 0.1 at different times T = 0.1, 0.2, 0.3. It has been observed that numerical solutions obtained by BDF-2 and BDF-3 provides better accuracy than BDF-1. The experiment was repeated for different values of viscosity ν at different time levels T. Results are tabulated and presented in Tables 2-4 for viscosity ν= 0.02, 0.01, 1 at different time levels T. Numerical schemes implementing BDF-1 and BDF-2 are respectively discussed in our papers [29] and [26]. In Table 5, we have compared the proposed numerical schemes using implicit solvers BDF-2 and BDF-3 with other schemes available in literature. It has been observed that the proposed numerical schemes shows excellent agreement with exact solutions as compared to the schemes given in [21] and [32]. Numerical results obtained by proposed schemes are also comparable to those given in [17]. Discrete root mean square error norm (L₂) and maximum error norm (L_∞) are computed for ν = 0.05, 0.0125, at different time levels and presented in Table 6.

Table 1

Numerical solutions obtained by proposed schemes and the exact solutions for Example 1: ν = 0.1, Δx = 0.0125 and Δt = 0.001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
0.1	0.25	5.3461E-01	5.3427E-01	5.3428E-01	5.3414E-01
	0.50	8.7741E-01	8.7740E-01	8.7740E-01	8.7728E-01
	0.75	7.6148E-01	7.6183E-01	7.6183E-01	7.6180E-01

0.2	0.25	4.2982E-01	4.2944E-01	4.2945E-01	4.2932E-01
	0.50	7.5423E-01	7.5403E-01	7.5403E-01	7.5381E-01
	0.75	7.4884E-01	7.4928E-01	7.4928E-01	7.4914E-01

0.3	0.25	3.5969E-01	3.5934E-01	3.5935E-01	3.5923E-01
	0.50	6.5227E-01	6.5197E-01	6.5197E-01	6.5173E-01
	0.75	6.9505E-01	6.9534E-01	6.9533E-01	6.9508E-01

Table 2

Numerical solutions obtained by proposed schemes and the exact solutions for Example 1: ν = 0.02, Δx = 0.0125 and Δt = 0.001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
	0.25	5.6316E-01	5.6293E-01	5.6293E-01	5.6277E-01
0.1	0.50	9.3943E-01	9.3965E-01	9.3964E-01	9.3954E-01
	0.75	8.4823E-01	8.4847E-01	8.4847E-01	8.4841E-01

	0.25	4.6318E-01	4.6285E-01	4.6285E-01	4.6266E-01
0.2	0.50	8.3868E-01	8.3868E-01	8.3868E-01	8.3844E-01
	0.75	9.3655E-01	9.3739E-01	9.3739E-01	9.3721E-01

	0.25	3.9201E-01	3.9167E-01	3.9167E-01	3.9150E-01
0.3	0.50	7.3916E-01	7.3894E-01	7.3894E-01	7.3864E-01
	0.75	9.3956E-01	9.4041E-01	9.4040E-01	9.4007E-01

Table 3

Numerical solutions obtained by proposed schemes and the exact solutions for Example 1: ν = 0.01, Δx = 0.0125 and Δt = 0.0001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
	0.25	1.8823E-01	1.8821E-01	1.8821E-01	1.8819E-01
1	0.50	3.7449E-01	3.7445E-01	3.7445E-01	3.7442E-01
	0.75	5.5615E-01	5.5610E-01	5.5610E-01	5.5605E-01

	0.25	1.0739E-01	1.0739E-01	1.0739E-01	1.0738E-01
2	0.50	2.1459E-01	2.1457E-01	2.1457E-01	2.1456E-01
	0.75	3.2133E-01	3.2131E-01	3.2131E-01	3.2128E-01

	0.25	7.5122E-02	7.5117E-02	7.5117E-02	7.5114E-02
3	0.50	1.5020E-01	1.5018E-01	1.5018E-01	1.5018E-01
	0.75	2.2487E-01	2.2485E-01	2.2485E-01	2.2481E-01

Table 4

Numerical solutions obtained by proposed schemes and the exact solutions for Example 1: ν = 1, Δx = 0.0125 and Δt = 0.0001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
	0.25	6.2911E-01	6.2905E-01	6.2905E-01	6.2904E-01
0.01	0.50	9.0577E-01	9.0573E-01	9.0573E-01	9.0571E-01
	0.75	6.5244E-01	6.5244E-01	6.5244E-01	6.5244E-01

	0.25	5.6322E-01	5.6313E-01	5.6313E-01	5.6312E-01
0.02	0.50	8.2020E-01	8.2012E-01	8.2012E-01	8.2010E-01
	0.75	5.9810E-01	5.9808E-01	5.9808E-01	5.9807E-01

	0.25	5.0650E-01	5.0639E-01	5.0639E-01	5.0639E-01
0.03	0.50	7.4273E-01	7.4262E-01	7.4262E-01	7.4259E-01
	0.75	5.4572E-01	5.4568E-01	5.4568E-01	5.4567E-01

Table 5

Comparison with different numerical methods and exact solution of Example 1 for ν = 0.01.

x	Time	[21]	[32]	[17]	BDF-2	BDF-3	Exact Solution
		Δt = 0.0001	Δt = 0.001	Δt = 0.01	Δt = 0.001	Δt = 0.001
	0.4	0.34819	0.34244	0.34229	0.34207	0.34208	0.34191
	0.6	0.27536	0.26905	0.26902	0.26908	0.26908	0.26896
0.25	0.8	0.22752	0.22145	.......	0.22157	0.22157	0.22148
	1.0	0.19375	0.18813	0.18817	0.18826	0.18826	0.18819
	3.0	0.07754	0.07509	0.07511	0.07513	0.07513	0.07511

	0.4	0.66543	0.67152	0.66797	0.66103	0.66103	0.66071
	0.6	0.53525	0.53406	0.53211	0.52969	0.52969	0.52942
0.50	0.8	0.44256	0.44143	.......	0.43936	0.43936	0.43914
	1.0	0.38047	0.37568	0.37500	0.37459	0.37460	0.37442
	3.0	0.15362	0.15020	0.15018	0.15022	0.15022	0.15018

	0.4	0.91201	0.94675	0.93680	0.91064	0.91063	0.91026
	0.6	0.77132	0.78474	0.77724	0.76761	0.76761	0.76724
0.75	0.8	0.65254	0.65659	.......	0.64772	0.64772	0.64740
	1.0	0.56157	0.56135	0.55833	0.55632	0.55632	0.55605
	3.0	0.22874	0.22502	0.22485	0.22491	0.22491	0.22481

Table 6

Comparison of Errors in L₂ norm and L_∞ norm with Δt = 0.0001 for different values of ν, corresponding to Example 1.

T	ν=0.05		ν=0.0125

	L₂	L_∞	L₂	L_∞
0.01	4.27E-06	1.84E-05	4.37E-06	1.38E-05
0.1	2.99E-05	4.83E-05	3.18E-05	4.97E-05
0.15	1.19E-04	1.86E-04	1.37E-04	2.27E-04
0.2	7.63E-05	1.26E-04	9.71E-05	1.87E-04
0.25	1.21E-04	2.32E-04	1.99E-04	5.17E-04

Figure 1, shows comparison between numerical and exact solutions of example 1 for viscosity ν= 1, 0.1 at different time levels T. It has been observed that numerical solutions using BDF-2 agrees exactly with exact solutions. Physical behavior of numerical solutions at different times T in 3D and contour form for ν = 0.0066 has been presented in Figure 2.

Fig. 1

Example 1: Numerical solutions at different times for Δx = 0.0125 and different values of ν and Δt, (a) ν = 1, Δt = 0.0001 (b) ν = 0.1, Δt = 0.001.

Fig. 2

Example 1: Physical behavior of numerical solutions at different times in 3D and contour for Δx = 0.0125, ν = 0.0066, Δt = 0.001(a) T = 0.1 (b) T = 2.

Most of the numerical methods available in literature fails to capture physical behavior of the equation as viscosity tends to zero (ν → 0). Figure 3 shows the proposed scheme is able to capture the physical behavior for ν = 0.0001, 0.00001. Hence the proposed schemes overcome the drawback of several other numerical schemes given in the literature.

Fig. 3

Example 1: Physical behavior of numerical solutions at different times with Δx = 0.0125 and Δt = 0.001. (a) ν = 0.0001 (b) ν = 0.00001.

Example-2

Next, we consider the Burgers equation [8]

ut+uux=νuxx,x∈[0,1]andt∈[0,T]

with initial condition u(x,0) = 4x(1 – x), 0 ≤ x ≤ 1, and the following boundary conditions u(0,t) = 0 = u(1,t), 0 ≤ t ≤ T.

The exact solution of the problem is

u(x,t)=2πν∑n=1∞Dnexp−n2π2νtnsin⁡(nπx)D0+∑n=1∞Dnexp−n2π2νtcos⁡(nπx)

where the Fourier coefficients D₀ and D_n are the following D0=∫01exp⁡{−Re3[x2(3−2x)]}dx,Dn=∫01exp⁡{−Re3[x2(3−2x)]}cos⁡(nπx)dx..

For example 2, we have compared solutions obtained by proposed methods with exact solutions for viscosity ν = 0.05 at different final time levels T = 0.1,0.2,0.3. The values are tabulated and presented in Table 7. We have repeated the experiment with ν= 0.0125 at time levels T = 1,2,3 and presented in Table 8. It has been observed that numerical solutions agrees with the exact solution. Moreover, numerical schemes using BDF-2 and BDF-3 provide more accurate solutions as compared to BDF-1. To validate the accuracy of proposed numerical schemes, we have compared it with existing numerical schemes [15], [21] and [17] for viscosity ν = 0.01. Table 9, shows that the proposed numerical schemes are comparable with the scheme given in [15] and it is better than the schemes given in [21] and [17]. Error norms (L₂) and (L_∞) corresponding to example 2 are tabulated for ν = 0.1, 0.02 in Table 10.

Table 7

Numerical solutions obtained by proposed schemes and the exact solutions for Example 2: ν = 0.05, Δx = 0.0125 and Δt = 0.0001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
	0.25	5.8696E-01	5.8693E-01	5.8693E-01	5.8690E-01
0.1	0.50	9.2821E-01	9.2823E-01	9.2823E-01	9.2821E-01
	0.75	8.4401E-01	8.4404E-01	8.4404E-01	8.4400E-01

	0.25	4.7481E-01	4.7477E-01	4.7477E-01	4.7473E-01
0.2	0.50	8.2664E-01	8.2664E-01	8.2664E-01	8.2659E-01
	0.75	8.8488E-01	8.8493E-01	8.8493E-01	8.8479E-01

	0.25	3.9750E-01	3.9746E-01	3.9746E-01	3.9736E-01
0.3	0.50	7.2861E-01	7.2859E-01	7.2859E-01	7.2843E-01
	0.75	8.7081E-01	8.7087E-01	8.7087E-01	8.7058E-01

Table 8

Numerical solutions obtained by proposed schemes and the exact solutions for Example 2: ν = 0.0125, Δx = 0.0125 and Δt = 0.0001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
	0.25	1.9410E-01	1.9408E-01	1.9409E-01	1.9406E-01
1	0.50	3.8460E-01	3.8456E-01	3.8459E-01	3.8453E-01
	0.75	5.6774E-01	5.6770E-01	5.6774E-01	5.6766E-01

	0.25	1.0927E-01	1.0926E-01	1.0927E-01	1.0926E-01
2	0.50	2.1818E-01	2.1816E-01	2.1817E-01	2.1815E-01
	0.75	3.2622E-01	3.2619E-01	3.2620E-01	3.2615E-01

	0.25	7.6028E-02	7.6023E-02	7.6025E-02	7.6020E-02
3	0.50	1.5197E-01	1.5196E-01	1.5196E-01	1.5195E-01
	0.75	2.2640E-01	2.2639E-01	2.2640E-01	2.2631E-01

Table 9

Comparison with different numerical methods and exact solution of example 2 for ν = 0.01.

x	Time	[21]	[17]	[15]	BDF-2	BDF-3	Exact Solution
		Δt = 0.0001	Δt = 0.01	Δt = 0.001	Δt = 0.001	Δt = 0.001
	0.4	0.36911	0.36273	0.36217	0.36244	0.36245	0.36226
0.25	0.6	0.28905	0.28212	0.28197	0.28217	0.28217	0.28204
	0.8	0.23703	...	0.23040	0.23055	0.23055	0.23045
	1.0	0.20069	0.19467	0.19465	0.19477	0.19477	0.19469
	3.0	0.07865	0.07613	0.07613	0.07615	0.07615	0.07613

	0.4	0.68818	0.69186	0.68357	0.68398	0.68398	0.68368
0.50	0.6	0.55425	0.55125	0.54822	0.54859	0.54860	0.54832
	0.8	0.46011	...	0.45363	0.45394	0.45395	0.45371
	1.0	0.39206	0.38627	0.38561	0.38586	0.38586	0.38568
	3.0	0.15576	0.15218	0.15217	0.15222	0.15222	0.15218

	0.4	0.92194	0.94940	0.92050	0.92079	0.92078	0.92050
0.75	0.6	0.78676	0.79399	0.78293	0.78333	0.78333	0.78299
	0.8	0.66777	...	0.66264	0.66303	0.66303	0.66272
	1.0	0.57491	0.57170	0.56924	0.56959	0.56959	0.56932
	3.0	0.23183	0.22778	0.22774	0.22784	0.22784	0.22774

Table 10

Comparison of Errors in L₂ norm and L_∞ norm with Δt = 0.001 for different values of ν, corresponding to Example 2.

T	ν=0.1		ν=0.02

	L₂	L_∞	L₂	L_∞
0.01	9.83E-05	5.44E-04	3.88E-05	2.90E-04
0.05	8.29E-05	2.83E-04	7.72E-05	2.39E-04
0.1	9.53E-05	1.79E-04	1.37E-04	2.92E-04
0.15	1.18E-04	1.80E-04	2.14E-04	5.27E-04
0.2	1.43E-04	1.99E-04	3.22E-04	8.93E-04

Figure 4, shows that numerical solutions for test problem 2 agrees exactly with analytic solutions at each nodal points for viscosity ν = 0.02,0.01. The physical behavior of computed solutions is depicted in Figure 5 through contour and surface plots for viscosity ν = 0.05 and at different time levels T. Figure 6 shows the proposed scheme is able to capture the physical behavior for ν = 0.0001,0.00001.

Fig. 4

Example 2: Numerical solutions at different times for Δx = 0.0125 and different values of ν and Δt, (a) ν = 0.02, Δt = 0.0001 (b) ν = 0.01, Δt = 0.001.

Fig. 5

Example 2: Physical behavior of numerical solutions at different times in 3D and contour for Δx = 0.0125, ν = 0.05, Δt = 0.001 (a) T = 0.1 (b) T = 2.

Fig. 6

Example 2: Physical behavior of numerical solutions at different times with Δx = 0.0125 and Δt = 0.001. (a) ν = 0.0001 (b) ν = 0.00001.

Example-3

We also consider Burgers equation [39]

ut+uux=νuxx,x∈[0,1]andt∈[0,T]

with boundary conditions

u(0,t)=0=u(1,t),0≤t≤T,

and initial condition

u(x,0)=2νπsin⁡(πx)2+cos⁡(πx),0≤x≤1.

The exact solution for this problem is

u(x,t)=2νπexp⁡(−π2νt)sin⁡(πx)2+exp⁡(−π2νt)cos⁡(πx)

In our first computation we have taken ν = 0.02, Δx = 0.0125, Δt = 0.001 and compared the results obtained by proposed schemes with the exact solutions and reported in Table 11. In our next computation we took nu = 0.00667 and the numerical results are tabulated in Table 12. From Tables 11 and 12 we infer that numerical solutions obtained from BDF-2 and BDF-3 lies close to exact solutions. Next, we compared our numerical results with schemes given in Asaithambi [1] and Mittal and Jain [24] for ν = 0.1, 0.5 and presented in Tables 13 and 14. Once again the proposed numerical schemes provide better accuracy than the schemes given in [1] and [24].

Table 11

Numerical solutions obtained by proposed schemes and the exact solutions for Example 3: ν = 0.02, Δx = 0.0125 and Δt = 0.001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
	0.25	2.8267E-02	2.8266E-02	2.8266E-02	2.8267E-02
1	0.50	5.1579E-02	5.1578E-02	5.1578E-02	5.1577E-02
	0.75	5.1393E-02	5.1391E-02	5.1391E-02	5.1383E-02

	0.25	2.4178E-02	2.4178E-02	2.4178E-02	2.4177E-02
2	0.50	4.2343E-02	4.2341E-02	4.2341E-02	4.2338E-02
	0.75	3.9312E-02	3.9309E-02	3.9309E-02	3.9300E-02

	0.25	2.0557E-02	2.0556E-02	2.0556E-02	2.0555E-02
3	0.50	3.4760E-02	3.4758E-02	3.4758E-02	3.4754E-02
	0.75	3.0560E-02	3.0557E-02	3.0557E-02	3.0549E-02

Table 12

Numerical solutions obtained by proposed schemes and the exact solutions for Example 3: ν = 0.00667, Δx = 0.0125 and Δt = 0.001.

T	x	Computed Solution			Exact Solution

		BDF-1	BDF-2	BDF-3
	0.25	1.04178E-02	1.04178E-02	1.04178E-02	1.04178E-02
1	0.50	1.96104E-02	1.96103E-02	1.96103E-02	1.96103E-02
	0.75	2.07301E-02	2.07300E-02	2.07300E-02	2.07285E-02

	0.25	9.91134E-03	9.91132E-03	9.91132E-03	9.91140E-03
2	0.50	1.83618E-02	1.83618E-02	1.83618E-02	1.83615E-02
	0.75	1.88182E-02	1.88180E-02	1.88180E-02	1.88156E-02

	0.25	9.42216E-03	9.42213E-03	9.42213E-03	9.42222E-03
3	0.50	1.71928E-02	1.71927E-02	1.71927E-02	1.71922E-02
	0.75	1.71307E-02	1.71304E-02	1.71304E-02	1.71275E-02

Table 13

Comparison with different numerical methods and exact solution of Example 3: ν = 0.1, T = 0.001, Δx = 0.0001.

x	[24]	[1]	BDF-2	BDF-3	Exact Solution
0.1	0.065750	0.065750	0.065750	0.065750	0.065750
0.2	0.131383	0.131383	0.131383	0.131383	0.131383
0.3	0.196281	0.196281	0.196281	0.196281	0.196281
0.4	0.258576	0.258576	0.258576	0.258576	0.258576
0.5	0.313850	0.313850	0.313849	0.313849	0.313849
0.6	0.352972	0.352972	0.352972	0.352972	0.352972
0.7	0.359443	0.359444	0.359443	0.359443	0.359443
0.8	0.309579	0.309583	0.309581	0.309581	0.309580
0.9	0.184751	0.184756	0.184754	0.184754	0.184754

Table 14

Comparison with different numerical methods and exact solution of Example 3: ν = 0.5, T = 0.001, Δt = 0.0001.

x	[24]	[1]	BDF-2	BDF-3	Exact Solution
0.1	0.327870	0.327874	0.327869	0.327869	0.327870
0.2	0.655071	0.655078	0.655069	0.655069	0.655069
0.3	0.978416	0.978420	0.978413	0.978413	0.978413
0.4	1.288469	1.288485	1.288464	1.288464	1.288464
0.5	1.563074	1.563096	1.563065	1.563065	1.563064
0.6	1.756654	1.756691	1.756645	1.756646	1.756642
0.7	1.787204	1.787281	1.787214	1.787215	1.787206
0.8	1.537649	1.537794	1.537707	1.537708	1.537694
0.9	0.916786	0.916941	0.916871	0.916871	0.916860

Example-4

In this test example, consider Burgers equation [25]

ut+uux=νuxx,x∈[0,5]andt∈[0,T]

with the boundary conditions

u(0,t)=0=u(5,t),0≤t≤T,

and initial condition

u(x,0)=sin(πx),0<x<1−12sin(πx),1≤x<20,2≤x<5

We have presented two dimensional figures to show the physical behavior of the above problem. In our computations we have considered viscosity, ν = 0.1, 0.01, Δt = 0.001 and Δx = 0.05. In Figures 7 and 8 numerical solutions corresponding to T ≤ 5 have been presented. Solution profiles indicate that the numerical solutions generated by proposed methods exhibit true physical behavior and is similar to those given in Mittal and Singhal [25] and Mittal and Jain [24].

Fig. 7

Example 4: Solution profiles for T ≤ 5, ν = 0.1, Δt = 0.001 and Δx = 0.05.

Fig 8

Example 4: Solution profiles for T ≤ 5, ν = 0.01, Δt = 0.001 and Δx = 0.05

6 Conclusion

In this paper, Burgers equation is solved by semidiscretization technique in combination with backward differentiation formulas. The proposed schemes are tested on four test problems and numerical solutions have been compared with exact solution. Numerical solutions shows excellent agreement with the exact solutions. It has been observed that numerical schemes with implicit solver BDF-2 and BDF-3 provide better solutions than BDF-1. The proposed schemes are comparable with the scheme given in [15], [17] and it is better than the schemes given in [21], [32], [1] and [24]. Hence the proposed numerical schemes are efficient and reliable for solving the Burgers equation even for small value of viscosity parameter. Moreover, thescheme can be extended to solve other nonlinear problems like modified Burgers equation and Burgers equation in two dimension.

Acknowledgement

The authors are very thankful to the reviewers for their valuable comments and suggestions.

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Received: 2017-03-07

Revised: 2017-10-22

Accepted: 2017-12-09

Published Online: 2018-06-02

Published in Print: 2018-09-25

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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https://doi.org/10.1515/nleng-2017-0068

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Burgers equation; Method of lines; Backward Differentiation Formula; Stability

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