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High-accuracy quasi-variable mesh method for the system of 1D quasi-linear parabolic partial differential equations based on off-step spline in compression approximations
- RK Mohanty^{1}Email author and
- Sachin Sharma^{2}
https://doi.org/10.1186/s13662-017-1274-3
© The Author(s) 2017
Received: 9 May 2017
Accepted: 10 July 2017
Published: 1 August 2017
Abstract
In this article, we propose a new two-level implicit method of accuracy two in time and three in space based on spline in compression approximations using two off-step points and a central point on a quasi-variable mesh for the numerical solution of the system of 1D quasi-linear parabolic partial differential equations. The new method is derived directly from the continuity condition of the first-order derivative of the spline function. The stability analysis for a model problem is discussed. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, we solve the generalized Burgers-Fisher equation, generalized Burgers-Huxley equation, coupled Burgers-equations and heat equation in polar coordinates. We demonstrate that the proposed method enables us to obtain high accurate solution for high Reynolds number.
Keywords
MSC
1 Introduction
The quasi-linear parabolic equation describes a wide class of physical phenomenon such as the interaction between reaction mechanism, convection, effects and diffusion transports. It is used in many fields such as chemistry, biology, metallurgy and engineering. The one-dimensional viscous generalized Burgers-Fisher equation (GBFE) and generalized Burgers-Huxley equation (GBHE) are famous examples of quasi-linear parabolic equations.
In both cases, equations describe the interaction between diffusion, convection and reaction.
The GBFE has wide applications in the fields such as gas dynamics, fluid mechanics, elasticity, heat conduction and plasma physics. The well-known equation (1.4) was first used by Fisher [1] to describe the propagation of gene in a habitat. In his memory, it is generally referred as Fisher’s equation. When \(\alpha = 0\) and \(\delta = 1\), equation (1.4) reduces to the classical Fisher equation. Kolmogorov et al. [2] independently wrote down the same equation to describe the dynamic spread of a combustion front. This equation has been found in various contexts in which a perturbation spreads in an excitable medium.
The GBHE was investigated by Satsuma et al. [3] in 1987. When \(\varepsilon = 1\), \(\alpha = 0\), \(\delta = 1\), equation (1.5) reduces to the Huxley equation and describes nerve pulse propagation in nerve fibers and wall motion in liquid crystals. For \(\varepsilon = 1\), \(\beta = 0\), equation (1.5) reduces to the generalized Burgers equation, which describes the far field of wave propagation in nonlinear dissipative systems. When \(\varepsilon = 1\), \(\alpha = 0\), \(\beta = 1\) and \(\delta = 1\), equation (1.5) becomes the Fitz-Hugh-Nagumo (FHN) equation which is the reaction diffusion equation used in circuit theory and biology. When \(\alpha \ne 0\), \(\beta \ne 0\) and \(\delta = 1\), equation (1.5) turns into the Burgers-Huxley equation (BHE) and shows a prototype model for describing the interaction between diffusion transports, convection and reaction mechanisms.
There has been vast variety of numerical methods, such as finite element methods, finite difference methods, spectral techniques and finite volume methods for quasi-linear parabolic initial-boundary value problems. In recent years, various numerical methods were used by the researchers to solve GBHE and GBFE. A fourth-order scheme for GBHE was proposed by Bratsos [4]. Mohammadi [5] has discussed a spline method for GBFE. Zhang et al. [6] solved GBFE using the local discontinuous Galerkin method. Diaz [7] analysed the solitary wave solution of the BHE through Cardano’s method. Mittal and Tripathi [8] discussed the schemes using collocation of cubic B-splines for numerical solutions of GBFE and GBHE. A two-level implicit compact operator method of order two in time and four in space was discussed for the approximate solution of time dependent BHE by Mohanty et al. [9].
Higher-order finite difference methods on a uniform mesh for the solution of nonlinear parabolic equations were proposed by Jain et al. [10]. Mittal and Jiwari [11] developed differential quadrature method for numerical solution of coupled viscous Burgers’ equations. Mohanty et al. [12] used compact operator technique to solve coupled Burgers’ equations. In recent past, Talwar et al. [13] proposed spline in compression method based on three full-step grid points for the solution of 1D quasi-linear parabolic equations and in those methods the consistency equation is only second-order accurate and the method is not directly applicable to singular problems, which is a main drawback of those methods. To the best of the authors’ knowledge, no numerical method of order two in time and three in space, directly obtained from the consistency condition, for the solution of parabolic equation (1.1) on a quasi-variable mesh has been discussed in the literature so far.
In this paper, using a central point and two off-step points in x-direction, we propose a new two-level implicit method of accuracy two in time and three in space, based on spline in compression approximations for the solution of differential equation (1.1). The proposed method is obtained directly from the consistency condition and is of order three in space. Difficulties were experienced in the past for the higher-order spline solution of parabolic equation in polar coordinates. The solution usually deteriorates in the vicinity of the singularity. A special technique is required to handle such problems, whereas the proposed method is directly applicable to solve singular problems without any modification, which is the main attraction of our work. Our paper is arranged as follows: In Section 2, we discuss the non-polynomial spline in compression function and its properties on a quasi-variable mesh. In Section 3, we give derivation of the method. In Section 4, we generalize the proposed method for the system of quasi-linear parabolic PDEs. Stability analysis for model problem is discussed and it is shown that the linear scheme is unconditionally stable in Section 5. In this section, we also discuss the stability analysis for a fourth-order parabolic equation which is consistent with system of 1D quasi-linear parabolic PDEs. In Section 6, numerical results are presented for some benchmark problems with tabular and graphical illustrations and compare the results with the results obtained by other researchers. Final remarks are given in Section 7.
2 Spline in compression approximations and its properties
For the approximate solution of the initial-boundary value problems (1.1)-(1.3), we discretize the space interval \([0,1]\) as \(0 = x_{0} < x_{1} <\cdots < x_{N} < x_{N + 1} = 1\), where N is a positive integer. The spline approximation consists of two off-step points \(x_{l \pm 1/2}\) and a central point \(x_{l}\), \(l = 0,1,2,\ldots,N\) with two end points \(x_{0}\) and \(x_{N + 1}\), where \(h_{l} = x_{l} - x_{l - 1}\), \(l = 1,2,\ldots,N + 1\), be the mesh size in x-direction and \(k = t_{j + 1} - t_{j} > 0\), \(j = 0,1,2,\ldots \) be the mesh spacing in t-direction. Spatial grid points are defined by \(x_{l} = x_{0} + \sum_{i = 1}^{l} h_{i}\), \(l = 1(1)N + 1\), and the time steps are given by \(t_{j} = jk\), \(j = 0(1)J\), where J is a positive integer. The mesh ratio is denoted by \(\sigma_{l} = (h_{l + 1}/h_{l}) > 0\), \(l = 1(1)N\). The neighboring off-step points are defined as \(x_{l - 1/2} = x_{l} - \frac{h_{l}}{2}\) and \(x_{l + 1/2} = x_{l} + \frac{\sigma_{l}h_{l}}{2}\), \(l = 1(1)N\). For \(\sigma_{l} = 1\), it reduces to the uniform mesh case. Let \(U_{l}^{j} = u(x_{l},t_{j})\) be the exact solution value of \(u(x,t)\) and is approximated by \(u_{l}^{j}\). For simplicity, we consider \(\sigma_{l} = \sigma\) (a constant ≠1), \(l = 1(1)N\). For \(\sigma>1\) or \(\sigma<1\), the mesh sizes are either increasing or decreasing in order. Such a mesh is called a quasi-variable mesh.
- (i)
\(P_{j}(x) \in C^{2}[0,1]\), and
- (ii)
\(P_{j}(x_{l}) = U_{l}^{j}\), \(P_{j}(x_{l - 1}) = U_{l - 1}^{j}\), where w is an arbitrary parameter and \(P''_{j}(x_{l}) = M_{l}^{j} = U_{xxl}^{j}\), \(P''_{j}(x_{l \pm 1}) = M_{l \pm 1}^{j} = U_{xxl \pm 1}^{j}\), \(P''_{j}(x_{l \pm 1/2}) = M_{l \pm 1/2}^{j} = U_{xxl \pm 1/2}^{j}\), \(l = 0(1)N + 1\), \(j > 0\).
The above equation has an infinite number of roots, the smallest positive non-zero root being given by \(\mu_{l} = \mu = 8.986818916\). When \(w \to 0\), then \(( \alpha_{l},\beta_{1l},\beta_{2l},\gamma_{l} ) \to ( \frac{\sigma}{3},\frac{\sigma}{6},\frac{1}{6},\frac{1}{3} )\), and equation (2.8) reduces to a cubic spline relation.
Equations (2.16) and (2.17) are two important properties of non-polynomial spline in compression function \(P_{j}(x)\).
3 Derivation of the numerical method
For the derivation of the method, we simply follow the approaches given by Mohanty [14].
The proposed non-polynomial spline in compression method (3.6) to be of \(O ( kh_{l} + h_{l}^{3} )\), the coefficients of \(kh_{l}^{2}\) and \(h_{l}^{4}\) in (3.42) must be zero.
Thus we obtain \(\theta = \frac{1}{2}\), \(a = \frac{ - (1 - \sigma + \sigma^{2})}{4}\), \(c = \frac{ - (1 - \sigma + \sigma^{2})}{2\sigma (1 + \sigma )}\) and the local truncation error given by (3.42) reduces to \(\hat{T}_{l}^{j} = O ( kh_{l}^{3} + h_{l}^{5} )\).
4 Method extended to a system of quasi-linear parabolic equations
5 Application and stability analysis
Note that the scheme (5.3) is of \(O(kh_{l} + h_{l}^{3})\) for the solution of differential equation (5.1) and is free from the terms \(1/(r_{l \pm 1})\), thus, it is very easily solved for \(l = 1(1)N\); \(j = 0,1,2,\ldots \) , in the solution region without any modification. We do not require any fictitious point to solve the singular problem.
Above inequality is satisfied for all values of η and, \(D(x) = \frac{ - \alpha \nu}{x}\), \(\alpha = 1\) and 2. Hence the scheme (5.14) is unconditionally stable.
The initial values of \(u, u_{t}\) are prescribed at \(t = 0\) and boundary values of \(u, u_{xx}\) are prescribed at \(x = 0\) and \(x = 1\). Since the grid lines are parallel to the coordinate axes and the values of \(u, u_{xx}\) are exactly known on the boundary, the values of successive tangential partial derivatives of \(u, u_{xx}\), i.e., the values of \(u_{t}\),\(u_{\mathit{xxt}}\), …, are also known on the boundary \(x = 0\) and \(x = 1\). Similarly, the values of \(u_{x}\), \(u_{xx}\), \(u_{tx}\), … are also known at \(t = 0\). Hence the values of \(u_{xx}(x,0) - u_{t}(x,0)\), \(u_{xx}(0,t) - u_{t}(0,t)\) and \(u_{xx}(1,t) - u_{t}(1,t)\) are known exactly on the boundary.
Both inequalities of (5.26) are true for all values of ψ. Hence the scheme (5.19a)-(5.19b) is unconditionally stable.
6 Numerical illustrations
In this section, we have solved several benchmark problems using the proposed method based on spline in compression and compared the results with the results obtained by other researchers. The exact solutions are provided in each case. The right hand side homogeneous functions, initial and boundary conditions are obtained using the exact solution as a test procedure. The linear equations are solved using a tri-diagonal solver, whereas nonlinear equations are solved using the Newton-Raphson method. While using the Newton-Raphson method, we choose 0 as the initial guess. All the computations are carried out using MATLAB codes.
The given interval \([0,1]\) is divided into \((N+1)\) parts with \(0 = x_{0} < x_{1} <\cdots<x_{N} < x_{N + 1} = 1\), where \(h_{l} = x_{l} - x_{l - 1}\), \(l = 1,2,\ldots,N + 1\) and \(\sigma = h_{l + 1}/h_{l} > 0\), \(l = 1,2,\ldots,N\).
By prescribing the total number of mesh points \((N+2)\), we can compute the value of \(h_{1}\) from (6.2) or (6.3). This is the first mesh spacing on the left and remaining mesh is determined by \(h_{l + 1} = \sigma h_{l}\), \(l = 1,2,\ldots,N\).
Example 1
This problem is solved with N= 10, 16, k = 0.0001 and mesh ratio \(\sigma = 0.9\) by present method. The following cases have been discussed for different values of the parameters \(\alpha, \beta, \gamma\) and δ, which are involved in equation (6.4).
Case 1.1: We choose \(\alpha = 0.001\), \(\beta = 0.001\).
Example 1 : Case 1.1(a)(i): Maximum absolute errors at \(\pmb{\delta = 1}\)
x | t | Method given in [ 23 ] | Method given in [ 5 ] | Proposed method ( 3.6 ) |
---|---|---|---|---|
0.1 | 0.001 | 1.11(–16) | 5.55(–16) | 5.55(–17) |
0.005 | 9.43(–16) | 1.77(–15) | 7.77(–16) | |
0.010 | 4.21(–15) | 2.55(–15) | 3.13(–14) | |
0.5 | 0.001 | 1.11(–16) | 3.88(–16) | 1.66(–16) |
0.005 | 4.44(–16) | 2.60(–15) | 1.38(–15) | |
0.010 | 1.66(–16) | 4.99(–15) | 8.16(–15) | |
0.9 | 0.001 | 0 | 1.05(–15) | 5.55(–17) |
0.005 | 1.99(–15) | 3.44(–15) | 0 | |
0.010 | 5.05(–15) | 5.16(–15) | 4.44(–16) |
Example 1 : Case 1.1 (a)(ii): Maximum absolute errors at \(\pmb{\delta = 4}\)
x | t | Method given in [ 23 ] | Method given in [ 5 ] | Proposed method ( 3.6 ) |
---|---|---|---|---|
0.1 | 0.001 | 1.11(–16) | 3.76(–14) | 1.11(–16) |
0.005 | 2.22(–16) | 1.43(–13) | 9.99(–16) | |
0.010 | 6.66(–16) | 2.39(–13) | 1.75(–16) | |
0.5 | 0.001 | 1.12(–16) | 3.20(–14) | 1.11(–16) |
0.005 | 3.33(–16) | 1.61(–13) | 3.33(–16) | |
0.010 | 3.33(–16) | 3.22(–13) | 2.88(–15) | |
0.9 | 0.001 | 1.11(–16) | 3.84(–14) | 1.11(–16) |
0.005 | 5.55(–16) | 1.45(–13) | 1.11(–16) | |
0.010 | 1.11(–15) | 2.41(–13) | 2.22(–16) |
Example 1 : Case 1.1(b): Maximum absolute errors
t | δ = 1 | δ = 4 | δ = 8 |
---|---|---|---|
1.0 | 3.44(–15) | 9.95(–14) | 9.55(–13) |
2.0 | 1.66(–15) | 9.61(–14) | 9.54(–13) |
3.0 | 1.33(–15) | 9.73(–14) | 9.61(–13) |
4.0 | 1.31(–15) | 9.76(–14) | 9.88(–13) |
5.0 | 1.22(–15) | 9.78(–14) | 9.89(–13) |
Example 1 : Case 1.2(a): Maximum absolute errors at \(\pmb{\delta = 1}\)
Example 1 : Case 1.2(b): Maximum absolute errors at \(\pmb{\delta = 2}\)
Example 1 : Case 1.2(c): Maximum absolute errors at \(\pmb{\delta = 4}\)
Example 1 : Case 1.3(a): Maximum absolute errors at \(\pmb{\delta = 1}\)
x | t | Method given in [ 25 ] | Method given in [ 5 ] | Proposed method ( 3.6 ) |
---|---|---|---|---|
0.1 | 0.5 | 1.68(–11) | 2.59(–12) | 5.66(–13) |
1.0 | 1.79(–11) | 2.74(–12) | 3.42(–13) | |
2.0 | 1.46(–11) | 2.74(–12) | 6.27(–13) | |
0.5 | 0.5 | 3.40(–12) | 7.36(–13) | 7.52(–14) |
1.0 | 3.72(–12) | 7.99(–13) | 4.81(–14) | |
2.0 | 3.13(–12) | 8.39(–13) | 3.06(–14) | |
0.9 | 0.5 | 1.31(–11) | 2.67(–12) | 5.55(–17) |
1.0 | 1.37(–11) | 2.96(–12) | 5.55(–17) | |
2.0 | 1.07(–11) | 3.24(–12) | 8.88(–16) |
Example 1 : Case 1.3(b): Maximum absolute errors at \(\pmb{\delta = 2}\)
x | t | Method given in [ 25 ] | Method given in [ 5 ] | Proposed method ( 3.6 ) |
---|---|---|---|---|
0.1 | 0.5 | 4.49(–11) | 2.83(–11) | 6.14(–12) |
1.0 | 4.19(–11) | 2.78(–11) | 7.28(–12) | |
2.0 | 2.70(–11) | 2.41(–11) | 9.17(–12) | |
0.5 | 0.5 | 8.13(–12) | 8.42(–12) | 2.24(–13) |
1.0 | 7.72(–12) | 8.50(–12) | 2.71(–13) | |
2.0 | 4.77(–12) | 7.85(–12) | 3.55(–13) | |
0.9 | 0.5 | 3.55(–11) | 3.13(–11) | 3.33(–16) |
1.0 | 3.23(–11) | 3.23(–11) | 3.33(–16) | |
2.0 | 1.98(–11) | 3.14(–11) | 4.44(–16) |
Example 4 : Case 1.3(c): Maximum absolute errors at \(\pmb{\delta = 8}\) .
x | t | Method given in [ 25 ] | Method given in [ 5 ] | Proposed method ( 3.6 ) |
---|---|---|---|---|
0.1 | 0.5 | 4.60(–11) | 9.37(–12) | 1.20(–11) |
1.0 | 4.39(–11) | 8.93(–12) | 1.25(–11) | |
2.0 | 3.78(–11) | 7.72(–12) | 1.31(–11) | |
0.5 | 0.5 | 7.03(–12) | 3.06(–12) | 4.36(–13) |
1.0 | 6.75(–12) | 2.99(–12) | 4.57(–13) | |
2.0 | 5.67(–12) | 2.74(–12) | 4.88(–13) | |
0.9 | 0.5 | 3.69(–11) | 1.19(–11) | 5.55(–16) |
1.0 | 3.45(–11) | 1.18(–11) | 6.66(–16) | |
2.0 | 2.84(–11) | 1.13(–11) | 6.66(–16) |
Example 2
Example 3
Example 4
The following cases have been discussed for different values of the parameters \(\alpha, \beta, \gamma\) and δ which are involved in Eq. (6.11).
Example 4 : Case 4.1: Maximum absolute errors
x | t | δ = 1 | δ = 2 | δ = 4 |
---|---|---|---|---|
0.1 | 0.1 | 6.7654(–17) | 6.3838(–16) | 6.4393(–15) |
0.5 | 4.6404(–17) | 4.8919(–16) | 4.6352(–15) | |
0.9 | 4.7054(–17) | 4.5797(–17) | 4.9682(–15) | |
0.5 | 0.1 | 2.9816(–17) | 3.4001(–16) | 3.2752(–15) |
0.5 | 4.2284(–17) | 4.3716(–16) | 4.2466(–15) | |
0.9 | 4.2609(–17) | 4.2674(–16) | 4.3854(–15) | |
0.9 | 0.1 | 1.0842(–19) | 6.9389(–18) | 2.7756(–17) |
0.5 | 5.4210(–19) | 6.9389(–18) | 5.5511(–17) | |
0.9 | 5.4210(–19) | 6.9389(–18) | 5.5511(–17) |
Example 4 : Case 4.2: Maximum absolute errors
x | t | β = 10 | β = 100 | β = 200 |
---|---|---|---|---|
0.1 | 0.1 | 6.4393(–15) | 6.6946(–14) | 6.7590(–13) |
0.5 | 2.6867(–14) | 4.4409(–14) | 2.6479(–14) | |
0.9 | 8.3267(–16) | 8.8818(–15) | 8.4044(–14) | |
0.5 | 0.1 | 5.2736(–15) | 4.6241(–14) | 4.5314(–13) |
0.5 | 9.9809(–14) | 7.0166(–14) | 6.7590(–13) | |
0.9 | 7.0499(–15) | 6.9666(–14) | 7.0732(–13) | |
0.9 | 0.1 | 1.1102(–16) | 7.7716(–16) | 7.6050(–15) |
0.5 | 2.6423(–14) | 1.3878(–15) | 1.4044(–14) | |
0.9 | 1.6653(–16) | 1.3878(–15) | 1.4044(–14) |
Example 4 : Case 4.3: Maximum absolute errors
x | t | \(\boldsymbol {\gamma = 10^{ - 2}}\) | \(\boldsymbol {\gamma = 10^{ - 3}}\) | \(\boldsymbol {\gamma = 10^{ - 4}}\) |
---|---|---|---|---|
0.1 | 0.1 | 1.1852(–13) | 6.4193(–14) | 7.1360(–15) |
0.5 | 1.1358(–13) | 3.6082(–14) | 1.0680(–15) | |
0.9 | 1.0858(–13) | 7.6883(–15) | 8.4099(–15) | |
0.5 | 0.1 | 8.8096(–14) | 5.2708(–14) | 6.9000(–14) |
0.5 | 8.4543(–14) | 7.0194(–14) | 6.9375(–14) | |
0.9 | 8.1157(–14) | 6.8168(–14) | 4.8003(–14) | |
0.9 | 0.1 | 2.3870(–15) | 8.8818(–16) | 1.4017(–15) |
0.5 | 2.2760(–15) | 1.3878(–15) | 1.4017(–15) | |
0.9 | 2.2204(–15) | 1.4155(–15) | 7.7716(–16) |
Example 4 : Case 4.4: Maximum absolute errors \(\pmb{\alpha = 1}\) , \(\pmb{\beta = 0}\)
N + 1 | Proposed method ( 3.6 ) | Method given in [ 4 ] | ||||
---|---|---|---|---|---|---|
\(\boldsymbol {R_{e} = 10^{2\vphantom{\sum^{i}}}}\) | \(\boldsymbol {R_{e} = 10^{4}}\) | \(\boldsymbol {R_{e} = 10^{6}}\) | \(\boldsymbol {R_{e} = 10^{2}}\) | \(\boldsymbol {R_{e} = 10^{4}}\) | \(\boldsymbol {R_{e} = 10^{6}}\) | |
8 | 8.8258(–06) | 1.9258(–09) | 1.0430(–13) | 4.1061(–04) | 8.3710(–08) | 8.4373(–12) |
16 | 4.9239(–07) | 9.7076(–11) | 1.0463(–14) | 1.1067(–04) | 2.2489(–08) | 2.2700(–12) |
32 | 4.2004(–08) | 7.3269(–12) | 9.9642(–16) | 2.7680(–05) | 5.7437(–09) | 5.7920(–13) |
64 | 1.0396(–08) | 1.2489(–12) | 1.3421(–16) | 6.9473(–06) | 1.4564(–09) | 1.4698(–13) |
Example 5
Example 6
Example 7
Here \(0 < \varepsilon \ll 1\) is the viscosity, \(R_{e} = \varepsilon^{ - 1} > 0\) is the Reynolds number, \(\alpha_{1}\) and \(\beta_{1}\) are real constants, \(\alpha_{2}\) and \(\beta_{2}\) are arbitrary constants depending on the system parameters [12]. The coupled Burgers equations (6.21) and (6.22) represent a system of one-space dimensional quasi-linear parabolic equations with two unknown variables u and v.
The values of parameters are given by \(\alpha_{1} = \beta_{1} = - 2\) and \(\alpha_{2} = \beta_{2} = 1\). The exact solutions of equations (6.21), (6.22) are \(u(x,t) = e^{ - t}\sin x\) and \(v(x,t) = e^{ - t}\sin x\) (see [22]).
Example 7 : Maximum absolute error at \(\pmb{\varepsilon = 1}\) , \(\pmb{\alpha_{1} = \beta_{1} = - 2}\) , \(\pmb{\alpha_{2} = \beta_{2} = 1}\) , \(\pmb{h = 2\pi /100}\) , \(\pmb{k = 0.01}\)
Example 7 : Maximum absolute errors, \(\pmb{\varepsilon = 1}\) , \(\pmb{\alpha_{1} = \beta_{1} = - 2}\) , \(\pmb{\alpha_{2} = \beta_{2} = 1}\) , \(\pmb{\lambda = 1.6/4\pi^{2}}\)
N + 1 | Proposed method ( 4.28 ) | Method given in [ 10 ] | |||
---|---|---|---|---|---|
t = 1.0 | t = 2.0 | t = 1.0 | t = 2.0 | ||
8 | u | 5.7826(–04) | 4.2579(–05) | 7.4756(–04) | 6.1386(–04) |
v | 5.7826(–04) | 4.2579(–05) | 7.4756(–04) | 6.1386(–04) | |
16 | u | 3.5478(–05) | 2.6105(–07) | 5.1124(–05) | 3.8618(–05) |
v | 3.5478(–05) | 2.6105(–07) | 5.1124(–05) | 3.8618(–05) | |
32 | u | 2.2070(–06) | 1.6238(–06) | 3.3516(–06) | 2.4173(–06) |
v | 2.2070(–06) | 1.6238(–06) | 3.3516(–06) | 2.4173(–06) | |
64 | u | 1.3777(–07) | 1.0137(–07) | 2.2035(–07) | 1.5114(–07) |
v | 1.3777(–07) | 1.0137(–07) | 2.2035(–07) | 1.5114(–07) | |
128 | u | 8.6048(–09) | 6.3339(–09) | 1.3837(–08) | 9.3878(–09) |
v | 8.6048(–09) | 6.3339(–09) | 1.3837(–08) | 9.3878(–09) |
Example 8
In this example, we choose a uniform mesh size to compute the numerical solution for different parameters \(\alpha_{1} = \beta_{1} = 2\) and \(\alpha_{2} = \beta_{2} = - 1\), the exact solutions of equations (6.21), (6.22) are \(u(x,t) = e^{ - \varepsilon \pi^{2}t}\cos (\pi x)\) and \(v(x,t) = e^{ - \varepsilon \pi^{2}t}\cos (\pi x)\).
Example 8 : Maximum absolute errors at \(\pmb{t = 1}\) , \(\pmb{\alpha_{1} = \beta_{1} = 2}\) , \(\pmb{\alpha_{2} = \beta_{2} = - 1}\) , \(\pmb{\lambda = 3.2}\)
N + 1 | Proposed method ( 4.28 ) | Method given in [ 10 ] | |||||
---|---|---|---|---|---|---|---|
\(\boldsymbol {R_{e} = 100}\) | \(\boldsymbol {R_{e} = 200}\) | \(\boldsymbol {R_{e} = 250}\) | \(\boldsymbol {R_{e} = 100}\) | \(\boldsymbol {R_{e} = 200}\) | \(\boldsymbol {R_{e} = 250}\) | ||
8 | u | 6.3075(–06) | 3.9152(–06) | 3.2823(–06) | 8.0563(–06) | 4.3982(–06) | 3.5838(–06) |
v | 6.3075(–06) | 3.9152 (–06) | 3.2837(–06) | 8.0563(–06) | 4.3982(–06) | 3.5838(–06) | |
16 | u | 4.1145(–07) | 2.4266(–07) | 2.0268(–07) | 5.0129(–07) | 2.8249(–07) | 2.2612(–07) |
v | 4.1145(–05) | 2.4266(–07) | 2.0268(–07) | 5.0129(–07) | 2.8249(–07) | 2.2612(–07) | |
32 | u | 2.5696(–08) | 1.5236(–08) | 1.2647(–08) | 3.1617(–08) | 1.7629(–08) | 1.4379(–08) |
v | 2.5696(–08) | 1.5236(–08) | 1.2647(–08) | 3.1617(–08) | 1.7629(–08) | 1.4379(–08) | |
64 | u | 1.6133(–09) | 9.5490(–10) | 7.9187(–10) | 1.9755(–09) | 1.1015(–09) | 8.9877(–10) |
v | 1.6133(–09) | 9.5490(–10) | 7.9187(–10) | 1.9755(–09) | 1.1015(–09) | 8.9877(–10) | |
128 | u | 1.0082(–10) | 5.9683(–11) | 4.9531(–11) | 1.2337(–10) | 6.8803(–11) | 5.6159(–11) |
v | 1.0082(–10) | 5.9683(–11) | 4.9531(–11) | 1.2337(–10) | 6.8803(–11) | 5.6159(–11) |
Example 9
Example 9 : Maximum absolute errors at \(\pmb{t = 1}\) , \(\pmb{k = 0.01}\)
N + 1 | p = 1 | p = 2 | ||
---|---|---|---|---|
\(\boldsymbol {R_{e} = 10}\) | \(\boldsymbol {R_{e} = 100}\) | \(\boldsymbol {R_{e} = 10}\) | \(\boldsymbol {R_{e} = 100}\) | |
50 | 1.6813(–06) | 4.6778(–06) | 1.5631(–06) | 4.6698(–06) |
60 | 1.4553(–06) | 4.6745(–06) | 1.3410(–06) | 4.6676(–06) |
70 | 1.2087(–06) | 4.6677(–06) | 1.1388(–06) | 4.6634(–06) |
80 | 9.1317(–07) | 4.6789(–06) | 8.5845(–07) | 4.6621(–06) |
90 | 6.1444(–07) | 4.6794(–06) | 5.4123(–07) | 4.6623(–06) |
7 Final remarks
In this article, we have presented a new two-level implicit method based on spline in compression approximations of accuracy \(O(kh_{l} + h_{l}^{3})\) for the numerical solution of quasi-linear parabolic partial differential equation in one spatial dimension. Mathematical formulation of the proposed scheme using three spatial grid points is discussed in detail. We have extended the proposed scheme to the system of quasi-linear parabolic equations. The stability analysis of the present numerical approach for one-dimensional linear convection-diffusion equation and fourth-order parabolic equation is presented. We have solved several benchmark problems using proposed method and it successfully provides highly accurate solutions in different settings of parameters. For different values of the parameters involved in GBFE, we have computed maximum absolute errors in Example 1 and we observe that our method is giving more accurate results than the results obtained by [5, 23–25]. In Examples 2 and 3, we have plotted the graphs (Figure 1(a)-(b) and Figure 2(a)-(f)) at different time levels and for different values of parameters in GBFE, which exhibits that the numerical diffusion is dominated with the increasing diffusion coefficient ε, whereas the reaction is gradually dominant with the increasing coefficient β. Similar patterns of graphs have been presented in [8, 18–20]. We have compared the computed numerical solutions with the exact solutions of GBHE in Example 4. Maximum absolute errors have been tabulated. The results obtained are quite good and competent with exact solution available in the literature. In Examples 5 and 6, we have plotted the graphs (Figure 3(a)-(c) and Figure 4(a)-(c)) at different time levels and different values of parameters involved in GBHE which describes that solution curves decreases to zero as time increases and for small value of ε, solution curves behave like a shocks waves. We have computed maximum absolute errors for coupled viscous nonlinear Burgers’ equation in Examples 7 and 8. On comparing the nature of computed solution with the computed solutions available in [10, 11], we obtained better results by our scheme. Also we have plotted graphs (Figure 5(a)-(c)) of exact versus numerical solution \(t = 1\). In Example 9, we have solved singular parabolic partial differential equation in polar coordinates and obtained maximum absolute errors for cylindrical and spherical case. At \(t = 1\), we have plotted graph (Figure 6) of numerical versus exact solution for Example 9. It can be observed that the approximate solution computed with our scheme and exact solutions are identical.
Declarations
Acknowledgements
The authors thank the reviewers for their valuable comments and suggestions, which substantially improved the standard of the paper. This research work is supported by CSIR-SRF, Grant No: 09/045(1161)/2012-EMR-I.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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