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International Journal for Computational Methods in Engineering Science and Mechanics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ucme20
A Collocation Method for Numerical Solutions of Coupled Burgers’ Equations a
R. C. Mittal & A. Tripathi
a
a
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India Accepted author version posted online: 18 Jun 2014.Published online: 19 Aug 2014.
Click for updates To cite this article: R. C. Mittal & A. Tripathi (2014) A Collocation Method for Numerical Solutions of Coupled Burgers’ Equations, International Journal for Computational Methods in Engineering Science and Mechanics, 15:5, 457-471, DOI: 10.1080/15502287.2014.929194 To link to this article: http://dx.doi.org/10.1080/15502287.2014.929194
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International Journal for Computational Methods in Engineering Science and Mechanics, 15:457–471, 2014 c Taylor & Francis Group, LLC Copyright ISSN: 1550-2287 print / 1550-2295 online DOI: 10.1080/15502287.2014.929194
A Collocation Method for Numerical Solutions of Coupled Burgers’ Equations R. C. Mittal and A. Tripathi
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Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India
and the boundary conditions In this paper, we propose a collocation–based numerical scheme to obtain approximate solutions of coupled Burgers’ equations. The scheme employs collocation of modified cubic B-spline functions. We have used modified cubic B-spline functions for unknown dependent variables u, v, and their derivatives w.r.t. space variable x. Collocation forms of the partial differential equations result in systems of first–order ordinary differential equations (ODEs). In this scheme, we did not use any transformation or linearization method to handle nonlinearity. The obtained system of ODEs has been solved by strong stability preserving the Runge-Kutta method. The proposed scheme needs less storage space and execution time. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in earlier studies. The scheme is simple as well as easy to implement. The scheme provides approximate solutions not only at the grid points, but also at any point in the solution range. Keywords
Coupled Burgers’ equations, Modified cubic B-spline functions, SSP-RK54 scheme, Collocation method
1. INTRODUCTION In this work, we are discussing the numerical solutions of coupled Burgers’ equations, proposed by Esipov [1] while studying the model of polydispersive sedimentation. These equations are described by nonlinear partial differential equations of the following form, ut = uxx − ηuux − p(uv)x , a < x < b, 0 ≤ t ≤ T (1) vt = vxx − ξ vvx − q(uv)x , a < x < b, 0 ≤ t ≤ T (2) with the initial conditions u(x, 0) = f1 (x),
v(x, 0) = f2 (x), a < x < b,
(3)
Address correspondence to Amit Tripathi, Dept. of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India. E-mail:
[email protected] Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/ucme.
u(a, t) = g0 (t), v(a, t) = g2 (t),
u(b, t) = g1 (t), 0 ≤ t ≤ T v(b, t) = g3 (t), 0 ≤ t ≤ T
(4) (5)
where η and ξ are real constants, p and q are arbitrary constants depending on the system parameters such as Peclet number, Stokes velocity of particles due to gravity, and Brownian diffusivity [2]. The coupled Burgers equations belong to an important class of basic flow equations [3]. Mathematical models of such equations describe many physical problems (please refer [4–9] and the references within). A great amount of research work has been carried out in recent years for the study of coupled linear and nonlinear initial/boundary value problems. Many numerical algorithms (e.g., Harmonic Differential Quadrature Finite differences coupled approach [10], Adomian and variational methods, conjugate filter approach [11], etc.) are available to obtain approximate solutions of coupled equations as well as other nonlinear differential equations. Jain et al. [12] proposed a difference scheme to obtain numerical solutions of two-dimensional unsteady NavierStokes equations. One-space Burgers’ equation in polar coordinates has been solved in [13] using a finite difference scheme. The variational iteration method has been presented for solving coupled Burgers’ equations by Abdou and Soliman [14]. Dehghan et al. [15] used the Adomain-Pade technique to solve these equations. Khater et al. [16] obtained approximate solutions of these equations using a cubic-spline collocation method. Mohanty and Jain [17] proposed a number of difference schemes to obtain numerical solutions of a system of nonlinear parabolic differential equations with mixed derivatives and variable coefficients. An application of the meshfree interpolation method for computing numerical solutions of coupled nonlinear partial differential equations has been proposed by Siraj-ul-Islam et al. [18]. Rashid and Ismail [19] proposed a pseudo-spectral method to find numerical solutions of coupled Burgers’ equations. The exact solutions of these equations have been obtained by Kaya [20] using the Adomian Decomposition method and by Soliman [21] using a modified extended tanh-function method.
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More recently, a two–level implicit discretization scheme has been proposed in [22] to solve singularly perturbed two-space dimensional nonlinear parabolic equations. In 2011, Mittal and Arora [23] investigated the solutions of the coupled Burgers’ equations by using collocation of B-spline functions. The order of convergence of the scheme is found to be approximately two. In 2012, Siraj-ul-Islam et al. [24] obtained the numerical solutions of the transient nonlinear coupled Burgers’ equations by collocation of local radial basis functions. A two-level implicit off-step discretization scheme has been presented by Mohanty and Setia [25] to solve a system of two space–dimensional quasi-linear parabolic partial differential equations. In the same year, Mittal and Jiwari [26] obtained the solutions of coupled Burgers’ equations by using the differential quadrature method. The numerical results obtained by them are found to be in good agreement with the exact solutions and with the numerical solutions available in the literature. In this work, we present a simple numerical scheme for solving coupled Burgers’ equations described by Equations (1) and (2). The scheme uses collocation of modified cubic B-spline functions over finite elements for spatial variable and its derivatives, which produces a system of first–order ordinary differential equations. We solve this system by using the SSP-RK54 method [27]. The approximate solutions have been computed without transforming the equations and without using linearization. The scheme is very simple and easy to implement. The structure of this paper is as follows: In Section 2, we describe cubic B-spline collocation method. In Section 3, the method is described by making use of modified cubic B-spline functions. The procedure of implementing the present method for Equations (1) and (2) is described in Section 3.1. The procedure to obtain initial values is also given in this section. In Section 4, numerical experiments and discussions are presented for different test problems with graphical and tabular illustrations. Conclusions of the proposed scheme are given in Section 5.
2. DESCRIPTION OF METHOD In this method, we express approximate solution as a linear combination of cubic B-spline basis functions over the concerned approximation space. Let us consider an interval [a, b] as a one–dimensional domain of interest with the following uniform partition.
solutions u(x, t) and v(x, t), respectively, in the form: U N (x, t) =
αj (t)Bj (x), a ≤ x ≤ b, t > 0.
(6)
βj (t)Bj (x), a ≤ x ≤ b, t > 0.
(7)
j =−1
V N (x, t) =
N+1 j =−1
where αj (t) and βj (t) are unknown time–dependent quantities to be determined from the boundary conditions and collocation form of the differential equations. The cubic B-spline function Bj (x) is given by: ⎧ (x − xj −2 )3 , x ∈ [xj −2 , xj −1 ) ⎪ ⎪ ⎪ ⎪ (x − xj −2 )3 − 4(x − xj −1 )3 , x ∈ [xj −1 , xj ) ⎨ 1 Bj (x) = 3 (xj +2 − x)3 − 4(xj +1 − x)3 , x ∈ [xj , xj +1 ) h ⎪ ⎪ x ∈ [xj +1 , xj +2 ) (xj +2 − x)3 , ⎪ ⎪ ⎩ 0, otherwise (8) where B−1 , B0 , B1 , ...., BN−1 , BN , BN+1 form a basis over the region a ≤ x ≤ b. Each cubic B-spline function covers four elements so that each element is covered by four cubic B-spline functions. At a particular knot xj , there exist only three cubic Bsplines, namely Bj −1 , Bj , Bj +1 , which have positive values. We present values of Bj (x) and its two successive derivatives over different knots in Table 1. At any time t and at a particular knot value xj , the approximate value U N (x, t) and its two derivatives can be obtained in terms of the time–dependent parameters αj (t) by using approximate solution (6) and cubic B-spline functions (8) as follows: ⎫ Uj = αj −1 + 4αj + αj +1 ⎪ ⎬ hUj = 3(αj +1 − αj −1 ) (9) ⎪ h2 Uj = 6(αj −1 − 2αj + αj +1 ) ⎭ The similar values for the approximate solution V N (x, t) and its derivatives can be obtained as those for the solution U N (x, t). 3. MODIFIED CUBIC B-SPLINES COLLOCATION METHOD In the present work, we are using modified cubic B-spline basis functions [28]. In the case of Dirichlet boundary conditions, TABLE 1 Cubic B-Spline functions and their derivatives at the knot values.
: a = x0 < x1 < x2 < ... < xN = b, where xj , j = 0, 1, 2, ...., N are called knots and h = (xj − xj −1 ) = (b − a)/N, j = 1, 2, ..., N is the differencing interval. Our numerical treatment for solving considered coupled Burgers’ equations using collocation of cubic B-splines is to find approximate solutions U N (x, t) and V N (x, t) to the exact
N+1
x Bj (x) Bj (x) Bj (x)
xj −2
xj −1
xj
xj +1
xj +2
0 0 0
1 3/ h 6/ h2
4 0 −12/ h2
1 −3/ h 6/ h2
0 0 0
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these functions provide a diagonally dominant system of differential equations. Modified cubic B-spline functions are given by: ⎫ B˜ 0 (x) = B0 (x) + 2B−1 (x) ⎪ ⎪ ⎪ ⎪ B˜ 1 (x) = B1 (x) − B−1 (x) ⎬ B˜ j (x) = Bj (x), f or j = 2, 3, ..., (N − 2). (10) ⎪ ⎪ B˜ N−1 (x) = BN−1 (x) − BN+1 (x) ⎪ ⎪ ⎭ B˜ N (x) = BN (x) + 2BN+1 (x), where B−1 , B0 , B1 , ..., BN−1 , BN , BN+1 are cubic B-spline functions w.r.t.a uniform partition a = x0 < x1 < ... < xN−1 = b of the solution domain a ≤ x ≤ b by the knots xi with differencing interval h = xi − xi−1 , i = 1, 2, ..., N. Now, let us assume the approximate solutions using the collocation of modified cubic B-spline basis functions in the following form: U (x, t) = N
N
αj (t)B˜ j (x), a ≤ x ≤ b, t > 0.
(11)
βj (t)B˜ j (x), a ≤ x ≤ b, t > 0.
(12)
j =0
V N (x, t) =
N
Using the approximate solution (11) and the modified functions (10), we get 6α0 (0) = g0 (0) α0 (0) + 4α1 (0) + α2 (0) = f1 (x1 ) ... ... αN−2 (0) + 4αN−1 (0) + αN (0) = f1 (xN−1 ) 6αN (0) = g1 (0). which can be written as Aα 0 = [g0 (0) f1 (x1 ) ... f1 (xN−1 ) g1 (0)]T ,
where α 0 = [α0 (0) α1 (0) ... αN−1 (0) αN (0)]T ≡ [α00 α10 ... 0 αN−1 αN0 ]T and
3.1 Initial Values From the conditions (3), (4) and with the considered discretization, at time t = 0, we have u(x0 , 0) u(x1 , 0)
= =
g0 (0) f1 (x1 )
......... ......... ......... ......... u(xN−1 , 0) = f1 (xN−1 ) u(xN , 0)
j =0
x=xj
=
g1 (0).
6 0 ⎢1 4 1 ⎢ ⎢ 1 4 ⎢ ⎢ A=⎢ ⎢ ⎢ ⎢ ⎢ ⎣
1 ...
... ...
... ... 1
... 4 1
1 4 1 0 6
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(14) .
(N+1)×(N+1)
The solution of this system (13) can be using the Thomas algorithm. In the case of solution V N (x, t) (described by (12)), initial values for the parameters βj (t), j = 0, 1, 2, ....., N, say β0 (0), β1 (0), ......, βN−1 (0) and βN (0), can be obtained, in a similar manner that has been accomplished for αj (t) s, by making use of Equations (3), (5), (10), and (12), respectively.
3.2 Intermediate Values Discretizing the interconnected equations (1) and (2) at the internal knots simultaneously and using the solutions (11), (12), and modified functions (10), we have N
˜ α (t) B (x) j j =0 j x=xj N N ˜ ˜ −η α (t) B (x) α = j j =0 j j =0 j (t)Bj (x) ⎪ x=x j ⎪ ⎪ N N N N ⎪ ˜ ˜ ˜ ˜ ⎩ −p α (t) B (x) β (t) B (x) + β (t) B (x) j j j j =0 j j =0 j j =0 j j =0 αj (t)Bj (x) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨
⎤ ⎡ N ⎣ α˙ j (t)B˜ j (x)⎦
⎤
⎡
j =0
To find these numerical solutions of coupled Burgers’ equations using a modified set of cubic B-spline basis functions B˜ j (x), j = 0, 1, 2, ..., N, we proceed as follows: Our numerical method for solving the equations using collocation of modified cubic B-spline functions is to find approximate solutions U N (x, t) and V N (x, t) as given in (11) and (12), where αj (t) and βj (t), j = 0, 1, 2, ...., N, are time–dependent parameters to be determined from the boundary conditions and collocation form of the differential equations.
(13)
(15) x=xj
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R. C. MITTAL AND A. TRIPATHI
and ⎡ ⎣
N
⎤ β˙ j (t)B˜ j (x)⎦
j =0
N
˜ β (t) B (x) j j =0 j x=xj N N ˜ ˜ −ξ β (t) B (x) β = (16) j j =0 j j =0 j (t)Bj (x) ⎪ x=x j ⎪ ⎪ N N N N ⎪ ˜ ˜ ˜ ˜ ⎩ −q j =0 αj (t)Bj (x) j =0 βj (t)Bj (x) + j =0 βj (t)Bj (x) j =0 αj (t)Bj (x) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨
x=xj
x=xj
where j = 0, 1, 2, ..., (N − 1), a < x < b, t > 0. Using (10), Equation (15) reduces to a system of ordinary differential equations of the form:
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Cu γ˙u = φu
(17)
γ˙u = [dα0 /dt dα1 /dt ... dαN−1 /dt dαN /dt]T ≡ [α˙0 α˙1 ... αN−1 ˙ α˙N ] , φu = [φ1 φ2 ... φN−1 ]T , with T
⎧ 6 ⎪ 2 (αj −1 + 2αj + αj +1 ) − η(αj −1 + 4αj + αj +1 ) ⎪ ⎨h 3 ∗ h (αj +1 − αj −1 ) (18) φj = 3 −p[(α ⎪ j −1 + 4αj + αj +1 ) ∗ h (βj +1 − βj −1 ) ⎪ ⎩ +(βj −1 + 4βj + βj +1 ) ∗ h3 (αj +1 − αj −1 )], and 1 4 ⎢ 1 ⎢ ⎢ Cu = ⎢ ⎢ ⎢ ⎣
1 4 ...
⎤ 1 ... ...
... 1
4 1 1 4 1
⎡
1 4 ⎢ 1 ⎢ ⎢ Cv = ⎢ ⎢ ⎢ ⎣
where
⎡
and
1 4 ...
⎤ 1 ... ...
... 1
(19)
(N−1)×(N+1)
Note that Cu is (N − 1) × (N + 1) tridiagonal matrix and φu is (N + 1) × 1 known column vector, which depends on boundary conditions. Similarly, using (10), Equation (16) reduces to a system of ordinary differential equations of the form: Cv γ˙v = ψv
(20)
where γ˙v = [dβ0 /dt dβ1 /dt ... dβN−1 /dt dβN /dt]T ˙ ≡ [β˙0 β˙1 ... βN−1 β˙N ]T , ψv = [ψ1 ψ2 ... ψN−1 ]T , with ⎧ 6 (β + 2βj + βj +1 ) − ξ (βj −1 + 4βj + βj +1 ) ⎪ ⎪ ⎪ h2 3j −1 ⎨ ∗ h (βj +1 − βj −1 ) ψj = (21) 3 ⎪ −q[(α ⎪ j −1 + 4αj + αj +1 ) ∗ h (βj +1 − βj −1 ) ⎪ ⎩ +(βj −1 + 4βj + βj +1 ) ∗ h3 (αj +1 − αj −1 )],
1 4 1
(22)
(N−1)×(N+1)
Note that systems obtained from (17) and (20) are interconnected and therefore should be treated simultaneously. Now, using conditions given in (4) and (5), at boundary knots we have (∂u/∂t)x=x0 = g˙ 0 (t), (∂u/∂t)x=xN = g˙ 1 (t), (∂v/∂t)x=x0 = g˙ 2 (t), (∂v/∂t)x=xN = g˙ 3 (t) But u(x, t) and v(x, t) are approximated by U N (x, t) and V N (x, t) as per expressions (11) and (12), therefore we have 6α˙ 0 = g˙ 0 (t), 6β˙ 0 = g˙ 2 (t),
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
4 1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
6α˙ N = g˙ 1 (t) 6β˙ N = g˙ 3 (t).
(23) (24)
Since equations in (17) and (23) arise from the discretization of the same time derivatives, combining them we have a system of ordinary differential equations, of the form: Aγ˙u = [g˙ 0 (t) φ1 φ2 ..... φN−1 g˙ 1 (t)]T
(25)
where A is matrix given by (14) and φj s, j = 1, 2, ..., (N − 1), are given by (18). Similarly, from (20) and (24) we obtain another system of ordinary differential equations, of the form: Aγ˙v = [g˙ 2 (t) ψ1 ψ2 ..... ψN−1 g˙ 3 (t)]T
(26)
where A is matrix again given by (14) and ψj s, j = 1, 2, ..., (N − 1), are given by (21). Now, we solve these interconnected systems (25) and (26) of ordinary differential equations simultaneously step by step by using the SSP-RK54 method [27]. First, we solve the system using a variant of Thomas algorithm only once at each time level t > 0, hence we obtain a system of first–order ordinary differential equations that can be solved by using the SSP-RK54 scheme. Once the vectors α(t) and β(t) have been determined at a specified time level, we can compute the solutions at any point in the solution range. Consequently, the approximate solutions U N (x, t) and V N (x, t) are completely known.
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It is easy to analyze the computational complexity of the present scheme. Complexity of computing initial values is dominated by the complexity of the Thomas algorithm that has been used to solve the system (13). Thomas algorithm uses (2N + 1) divisions, (3N ) subtractions, and (3N ) multiplications. Hence, in all it uses (8N + 1) simple arithmetic operations. Complexity of computing intermediate values for the next time step is dominated by complexity of the procedure that has been used to solve the system (25). This procedure uses the SSP-RK54 method with a variant of the Thomas algorithm. It uses (82N − 38) additions, (50N − 10) divisions, (80N − 50) subtractions, and (194N − 76) multiplications. Thus, in total it uses (406N − 176) simple arithmetic operations. Hence, we resolve that the computational complexity of the present scheme is O(N).
4. NUMERICAL EXPERIMENTS AND DISCUSSIONS In this section, we present numerical solutions obtained by the proposed scheme for the following test problems considered by the different researchers. We have tested the efficiency and adaptability of the proposed scheme for different values of the parameters in different cases. To compare and analyze the computed numerical solutions w.r.t. the exact solutions and the solutions already available in the literature, we demonstrate the results specifically for the same parameters as those there. We also report the execution time that has elapsed in computing the solutions in different cases. For describing errors, we consider the following error norms, say for the solution U N (x, t); L∞ (u) = max |u(xj , t) − U N (xj , t)| 0≤j ≤N
and
L2 (u) =
N j =0
|u(xj , t) − U N (xj , t)|2 . N 2 j =0 |u(xj , t)|
where u(xj , t) is the exact solution and U N (xj , t) is the corresponding approximate numerical solution obtained by the proposed scheme.
The numerical order of convergence R of the proposed scheme is calculated by the following formula 1) log Error(N Error(N2 ) (27) R= 2 log N N1 where Error(N1 ) and Error(N2 ) are the L∞ errors at number of partitions N and 2N , respectively. 4.1 Example 1 Consider the coupled Burgers’s equations (Eqs. (1) and (2)) with the following initial and boundary conditions: u(x, 0) = v(x, 0) = sin(x), −π ≤ x ≤ π, and u(−π, t) = u(π, t) = 0, 0 ≤ t ≤ T v(−π, t) = v(π, t) = 0, 0 ≤ t ≤ T For these conditions with parameters p = q = 1 and η = ξ = −2, the exact solution of the equation is u(x, t) = v(x, t) = e−t sin(x), as described by Kaya in [20]. We compute the numerical solutions for different values of the parameters η, ξ, p and q with different values of time step length t. In our first computation, we take p = 1, q = 1, t = 0.01, η = ξ = −2 and maximum absolute errors, for the obtained solutions are computed at different time levels from t = 0.5 to t = 3.0. The execution time in computing the results for t = 0.5 to t = 3.0 is 0.28 seconds. The corresponding results are presented in Table 2. In our next computation, we compute the maximum absolute errors and corresponding execution times at time level t = 1 for the parameters p = 1, q = 1, η = ξ = −2 with different decreasing values of t. The corresponding results are reported in Table 3. In both computations, the results are same for u(x, t) and v(x, t) because of symmetric initial and boundary conditions. In our next computation, we analyze the behavior of computed numerical solutions by plotting the solution profiles for different settings of parameters, specifically for those taken by Mittal and Jiwari in [26]. The corresponding graphical illustrations are presented in Figures 1–9. Mittal and Jiwari depicted similar patterns in [26] as those in Figures 1, 5,
TABLE 2 Maximum error norms for Example 1 (p = 1, q = 1, t = 0.01, N = 50, and η = ξ = −2) Mittal and Jiwari [26] t 0.5 1.0 2.0 3.0
Present Method
L∞ (u)
L∞ (v)
L∞ (u)
L∞ (v)
CPU time (in seconds)
1.51688e-04 1.83970e-04 1.35250e-04 7.46014e-05
1.51688e-04 1.83970e-04 1.35250e-04 7.46014e-05
1.103080984e-04 1.336880384e-04 9.818252567e-05 1.029870405e-05
1.103080984e-04 1.336880384e-04 9.818252567e-05 1.029870405e-05
0.005 0.009 0.020 0.028
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R. C. MITTAL AND A. TRIPATHI
TABLE 3 Convergence rates in time at t = 1 from maximum error norms for Example 1 (p = 1, q = 1, N = 50, and η = ξ = −2) Mittal and Jiwari [26] t
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0.0200 0.0100 0.0050 0.0020 0.0010 0.0005
Present Method
L∞ (u)
L∞ (v)
L∞ (u)
L∞ (v)
CPU time (in seconds)
3.70976e-03 1.84705e-03 9.21572e-04 4.60280e-04 1.83970e-04 9.19422e-05
3.70976e-03 1.84705e-03 9.21572e-04 4.60280e-04 1.83970e-04 9.19422e-05
1.336881168e-03 1.336880384e-04 1.336871098e-04 1.336581399e-04 1.335581399e-04 1.519766363e-05
1.336881168e-03 1.336880384e-04 1.336871098e-04 1.336581399e-04 1.335581399e-04 1.519766363e-05
0.004 0.015 0.031 0.046 0.093 0.171
and 6. In our next computation, we take p = 1, q = 1, t = 0.001, η = ξ = −2, and L2 and L∞ error norms are computed at different time levels for different sizes of space step length. The corresponding computed results are reported in Table 4. From all the computations, we resolve that obtained numerical solutions are competent with the solutions available in the literature [19, 23, 26] and are in good agreement with the exact solutions. The graphical illustrations capture the coupled nature and physical behavior of Equations (1) and (2) faithfully w.r.t. all the parameters. We resolve that the numerical solutions produced by the present scheme are consistent with the dynamics and physical behavior of the considered equations for different settings of the parameters. The order of convergence is calculated, by formula given by the expression 27, as per tabulations in Table 5 for the parameters p = 1, q = 1, t = 0.001 and η = ξ = −2. We can conclude that order of convergence is approximately equal to 2, as achieved by Mittal and Arora in [23].
FIG. 1. Computed solutions v of Example 1 for different time levels (p = 1, q = 1, t = 0.01, η = ξ = −2).
FIG. 2. Computed solutions v of Example 1 for different time levels (p, η and ξ are fixed.).
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SOLUTIONS OF COUPLED BURGERS’ EQUATIONS
FIG. 3. Computed solutions v of Example 1 for different time levels (q, η and ξ are fixed.).
4.2 Example 2 In this example, numerical solutions of considered coupled Burgers’ equations are obtained for η = ξ = 2 with different values of p and q at different time levels. The initial and boundary conditions are taken from the exact solution, which is given
by [21] as a0 (1 −tanh(A(x − 2At))), 2q−1 v(x, t) = a0 2p−1 − tanh(A(x − 2At)) ,
u(x, t) =
FIG. 4. Computed solutions v of Example 1 for different time levels (p, q and η or ξ are fixed w.r.t. Figure 1.).
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could not achieved much w.r.t. the scheme of Mittal and Jiwari [26], but still we have improved it. The strength of our scheme is that, regardless of providing solutions for every point of the domain, it possesses an easy and economical implementation. 4.3 Example 3 In this example, we compute numerical solutions of coupled Burgers’ equations (1) and (2) with the following initial conditions sin(2π x), 0 ≤ x ≤ 0.5, u(x, 0) = 0, 0.5 < x ≤ 1. 0, 0 ≤ x ≤ 0.5, v(x, 0) = −sin(2π x), 0.5 < x ≤ 1.
FIG. 5. Computed solutions v of Example 1 for different time levels (p = 1, q = 2, t = 0.01, η = 1, ξ = 2).
where
a0 = 0.5 and A = 0.5a0
4pq−1 2p−1
.
The numerical solutions have been computed for the domain x ∈ [−10, 10] with t = 0.01 and number of partitions as 20. The maximum absolute errors have been computed and compared in Tables 6 and 7 with those available in the literature [16, 19, 26]. We resolve that the present scheme provides more accurate numerical solutions in the comparison of schemes proposed in [16, 19] and [26]. Of course, in terms of accuracy, we
and zero boundary conditions. The solutions have been computed for the domain [0,1] with 50 partitions. We compute the numerical solutions for different values of the parameters p, q, η and ξ with time step length 0.00001. In our first computation, we take p = q = 10, η = ξ = 2 and compute the numerical solutions at different time levels, viz. t = 0.1, 0.2, 0.3, 0.4. The corresponding graphical illustrations are presented in Figures 10 and 11. Evolution of computed solutions in the vicinity of time t = 0 is shown in Figure 12. In our next computation, we take p = q = 100, η = ξ = 2 and compute solutions at t = 0.1, 0.2, 0.3, 0.4. We present the corresponding graphical illustrations in Figures 13 and 14. Similar patterns have been obtained by Mittal and Arora in [23]. We also compare the maximum values of computed solutions with those obtained by Mittal and Arora in [23]. The corresponding tabular illustrations
FIG. 6. Computed solutions v of Example 1 for different time levels (p, η and ξ are fixed).
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SOLUTIONS OF COUPLED BURGERS’ EQUATIONS
FIG. 7. Computed solutions v of Example 1 for different time levels (p, η and ξ are fixed.).
FIG. 8. Computed solutions v of Example 1 for different time levels (q, η and ξ are fixed.).
FIG. 9. Computed solutions v of Example 1 for different time levels.
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TABLE 4 L2 and L∞ error norms for u(x, t) for Example 1 (p = 1, q = 1, t = 0.001, η = ξ = −2) N = 100 (in 0.187 seconds)
N = 128 (in 0.265 seconds)
L2 (u)
L∞ (u)
L2 (u)
L∞ (u)
L2 (u)
L∞ (u)
3.290246751e-005 1.645015122e-004 3.289759636e-004
2.977138375e-005 9.97752107e-005 1.210234936e-004
3.018117223e-005 3.940889626e-005 2.808088795e-005
1.845607611e-005 6.185486538e-005 1.125890507e-005
.... .... 2.887e-05
.... .... 1.159e-05
t
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0.1 0.5 1.0
Rashid [19], N = 128
TABLE 5 Maximum error norms for u(x, t) for different space step lengths in Example 1 (p = 1, q = 1, t = 0.001, and η = ξ = −2) At t = 0.1 N 32 64 128
At t = 0.5
L∞ (u)
Order of conv.
Order of conv. by [23]
L∞ (u)
Order of conv.
Order of conv. by [23]
CPU time (in seconds)
3.097314121e-004 7.499956676e-005 1.845607611e-005
—– 2.046063538 2.022786403
—– 2.001 1.999
1.037384509e-004 2.513268852e-004 6.185486538e-005
—— 2.045313818 2.022606025
—— 2.005 2.001
0.046 0.078 0.171
TABLE 6 Maximum error norms for u(x, t) in Example 2 (N = 21, t = 0.01, and η = ξ = 2) t 0.5 1.0 3.0
p
q
0.1 0.3 0.1 0.3 0.1 0.1
0.3 0.03 0.3 0.03 0.3 0.03
Khater [26] L∞ (u)
Rashid [19] L∞ (u)
Mittal [16] L∞ (u)
Present Method L∞ (u)
1.44e-003 6.68e-004 1.27e-003 1.30e-003 ....... .......
9.619e-004 4.310e-004 1.153e-003 1.268e-003 ....... .......
4.173e-005 4.585e-005 8.275e-005 9.167e-005 2.408e-004 2.747e-004
4.189217417e-005 4.584830094e-005 8.269641708e-005 9.147335667e-005 2.401202768e-004 2.704203611e-004
TABLE 7 Maximum error norms for v(x,t) in Example 2 (N = 21, t = 0.01, and η = ξ = 2) t 0.5 1.0 3.0
p
q
0.1 0.3 0.1 0.3 0.1 0.1
0.3 0.03 0.3 0.03 0.3 0.03
Khater [26] L∞ (v)
Rashid [19] L∞ (v)
Mittal [16] L∞ (v)
Present Method L∞ (v)
5.42e-004 1.20e-003 1.29e-003 2.35e-003 ........ ........
3.332e-004 1.148e-003 1.162e-003 1.638e-003 ....... .......
5.418e-005 2.826e-005 1.074e-004 5.673e-005 3.119e-004 1.663e-004
9.094743099e-006 2.48218881e-005 1.696286567e-005 4.965329678e-005 4.505480184e-005 1.498311672e-005
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SOLUTIONS OF COUPLED BURGERS’ EQUATIONS
FIG. 10. Numerical solution u(x, t) of Example 3 at different time levels for p = q = 10.
FIG. 11. Numerical solution v(x, t) of Example 3 at different time levels for p = q = 10.
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FIG. 12. Evolution of computed solutions of Example 3 in the vicinity of t = 0 (p = q = 10 and η = ξ = 2).
FIG. 13. Numerical solution u(x, t) of Example 3 at different time levels for p = q = 100.
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SOLUTIONS OF COUPLED BURGERS’ EQUATIONS
FIG. 14. Numerical solution v(x,t) of Example 3 at different time levels for p = q = 100.
TABLE 8 Maximum values of u and v at different time levels for p = q = 10 Maximum value of u
Maximum value of v
t
Present Scheme
Mittal [23]
At point
Present Scheme
Mittal [23]
At point
CPU time (in seconds)
0.1 0.2 0.3 0.4
0.144491495800 0.052356151890 0.019318838080 0.007184856672
0.14456 0.05237 0.01932 0.00718
0.58 0.54 0.52 0.50
0.143141957500 0.047006446750 0.017260356430 0.006416614856
0.14306 0.04697 0.01725 0.00641
0.66 0.56 0.52 0.50
0.81 1.47 2.16 2.98
TABLE 9 Maximum values of u and v at different time levels for p = q = 100 Maximum value of u
Maximum value of v
t
Present Scheme
Mittal [23]
At point
Present Scheme
Mittal [23]
At point
CPU time (in seconds)
0.1 0.2 0.3 0.4
0.041682987260 0.014770415340 0.005337325631 0.001978065014
0.04175 0.01479 0.00534 0.00198
0.46 0.58 0.54 0.52
0.050737669860 0.010356602970 0.003517189432 0.001294450199
0.05065 0.01033 0.00350 0.00129
0.76 0.64 0.56 0.52
0.86 1.61 2.44 3.26
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FIG. 15. Solution profiles u(x, t) at different time levels with increasing values of convection coefficient.
are presented in Tables 8 and 9. The effect of increasing the value of convection coefficient is depicted in Figure 15 for the values 20, 200, and 2000. It can be concluded that solution decays to zero with increase in time and convection coefficient. 5. CONCLUSIONS We present a method to obtain numerical solutions of coupled Burgers’equations by using collocation of modified cubic B-spline functions. The method successfully provides very accurate solutions in different settings of parameters. The performance of the method has been measured over the considered test problems, either by calculating the different error norms for different time levels or by comparing the nature of computed solutions with nature of solutions available in the literature as well as dynamics of the differential equations, in different cases. The computed solutions and different errors for these solutions are given in tabular form for different time levels in different settings of parameters. The corresponding execution time durations have also been reported. The results demonstrate that error decreases with the increase in time. Maximum absolute errors have been computed and compared with those available in the literature. We conclude that the method provides convergent approximations and handles the equations very well in different cases. The results obtained are quite satisfactory and competent with the solutions available in the literature. The method can be used without any need of complex calculations rather simple and elementary operations. The method is economical, easy to implement, and reliable. It can be used as an alternative for a large class of similar nonlinear coupled equations. ACKNOWLEDGEMENT The authors are very thankful to the reviewers for their through reviews, valuable comments, and suggestions to improve the quality of this paper.
FUNDING The author Amit Tripathi thankfully acknowledges the financial assistance provided by MHRD India.
REFERENCES 1. S.E. Esipov, Coupled Burgers Equations: A Model of Polydispersive Sedimentation, Physical Review E, vol. 52(4), pp. 3711, 1995. 2. J. Nee, J. Duan, Limit Set of Trajectories of the Coupled Viscous Burgers’ Equations, Appl. Math. Lett., vol. 11(1), pp. 57–61, 1998. 3. S. N. Atluri, The Meshless Method (MLPG) for Domain & BIE Discretizations, Vol. 677, Tech Science Press Forsyth, 2004. 4. J. M. Burgers, A Mathematical Model Illustrating the Theory of Turbulence, Advances in Appl. Mech., vol. 1, pp. 171–199, 1948. 5. J. D. Cole, et al., On a Quasi-linear Parabolic Equation Occurring in Aerodynamics, Quart. Appl. Math., vol. 9(3), pp. 225–236, 1951. 6. J. Nee, J. Duanw, Limit Set of Trajectories of the Coupled Viscous Burgers’ Equations, Appl. Math. Lett., vol. 11(1), pp. 57–61, 1998. 7. A. Ali, S. ul Islam, S. Haq, A Computational Meshfree Technique for the Numerical Solution of the Two-Dimensional Coupled Burgers’ Equations, Int. J. Comp. Meth. in Eng. Sci. and Mech., vol. 10(5), pp. 406–422, 2009. 8. J. Perko, B. Sarler, Weight Function Shape Parameter Optimization in Meshless Methods for Non-uniform Grids, Comp. Modeling in Eng. and Sci., vol. 19(1), pp. 55, 2007. 9. W. Zhang, C. Zhang, G. Xi, An Explicit Chebyshev Pseudospectral Multigrid Method for Incompressible Navier–Stokes Equations, Computers & Fluids, vol. 39(1), pp. 178–188, 2010. ¨ Civalek, Harmonic Differential Quadrature-Finite Differences Coupled 10. O. Approaches for Geometrically Nonlinear Static and Dynamic Analysis of Rectangular Plates on Elastic Foundation, J. Sound and Vibration, vol. 294(4), pp. 966–980, 2006. 11. G. W. Wei, Y. Gu, Conjugate Filter Approach for Solving Burgers. Equation, J. Comp. and Appl. Math., vol. 149(2), pp. 439–456, 2002. 12. M. Jain, R. Jain, R. Mohantyw, The Numerical Solution of the TwoDimensional Unsteady Navier-Stokes Equations Using Fourth-Order Difference Method, Int. J. Comp. Math., vol. 39(1-2), pp. 125–134, 1991. 13. R. Mohanty, M. Jain, High Accuracy Difference Schemes for The System of Two Space Nonlinear Parabolic Differential Equations with Mixed Derivatives and Variable Coefficients, J. Comp. and Appl. Math., vol. 70(1), pp. 15–32, 1996.
Downloaded by [Indian Institute of Technology Roorkee] at 01:50 29 April 2015
SOLUTIONS OF COUPLED BURGERS’ EQUATIONS 14. M. A. Abdou, A. A. Soliman, Variational Iteration Method for Solving Burger’s and Coupled Burger’s Equations, J. Comp. and Appl. Math., vol. 181(2), pp. 245–251, 2005. 15. M. Dehghan, A. Hamidi, M. Shakourifar, The Solution of Coupled Burgers. Equations Using Adomian–Pade Technique, Appl. Math. and Comp., vol. 189(2), pp. 1034–1047, 2007. 16. A. H. Khater, R. S. Temsah, M. M. Hassan, A Chebyshev Spectral Collocation Method for Solving Burgers-type Equations, J. Comp. and Appl. Math., vol. 222(2), pp. 333–350, 2008. 17. R. Mohanty, An O (K2+ h4) Finite Difference Method for One-Space Burger’s Equation in Polar Coordinates, Numer. Meth. for Partial Diff. Eqs., vol. 12(5), pp. 579–583, 1996. 18. S.-U. Islam, S. Haq, M. Uddin, A Mesh Free Interpolation Method for the Numerical Solution of the Coupled Nonlinear Partial Differential Equations, Engineering Analysis with Boundary Elements, vol. 33, pp. 399–409, 2009. 19. A. Rashid, A. Ismail, A Fourier Pseudospectral Method for Solving Coupled Viscous Burgers Equations, Comp. Meth. in Appl. Math., vol. 9(4), pp. 412–420, 2009. 20. D. Kaya, An Explicit Solution of Coupled Viscous Burgers’ Equation by the Decomposition Method, Int. J. Math. and Math. Sci., vol. 27(11), pp. 675–680, 2001. 21. A. A. Soliman, The Modified Extended Tanh-Function Method for Solving Burgers-type Equations, Physica A: Statistical Mechanics and its Applications, vol. 361(2), pp. 394–404, 2006.
471
22. R. Mohanty, S. Singh, A New Two-level Implicit Discretization of o (k2+ kh2+ h4) for the Solution of Singularly Perturbed Two-space Dimensional Non-linear Parabolic Equations, J. Comp. Appl. Math., vol. 208(2), pp. 391–403, 2007. 23. R.C. Mittal, G. Arora, Numerical Solution of the Coupled Viscous Burgers Equation, Comm. in Nonlin. Sci. and Numer. Sim., vol. 16(3), pp. 1304–1313, 2011. ˇ 24. Sarler B. Siraj-ul-Islam, R. Vertnik, G. Kosec, Radial Basis Function Collocation Method for the Numerical Solution of the Two-dimensional Transient Nonlinear Couple Burgers Equations, Applied Mathematical Modelling, vol. 36(3), pp. 1148–1160, 2012. 25. R. Mohanty, N. Setia, A New High Accuracy Two-level Implicit off-step Discretization for the System of two Space Dimensional Quasi-linear Parabolic Partial Differential Equations, Appl. Math. and Comp., vol. 219(5), pp. 2680–2697, 2012. 26. R. C. Mittal, R. Jiwari, Differential Quadrature Method for Numerical Solution of Coupled Viscous Burgers Equations, Int. J. Comp. Meth. in Eng. Sci. and Mech., vol. 13(2), pp. 88–92, 2012. 27. S. Gottlieb, On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations, J. Scientific Computing, vol. 25(1), pp. 105–128, 2005. 28. R. C. Mittal, R. K. Jain, Numerical Solutions of Nonlinear Burgers Equation with Modified Cubic B-splines Collocation Method, Appl. Math. and Comp., vol. 218, pp. 7839–7855, 2012.
