The Space–Time Kernel-Based Numerical Method for Burgers’ Equations

Abstract

It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent PDEs are collocated in both space and time first, contrary to spatial discretization, and time stepping procedures for time integration are then applied. Physically one cannot in general expect that the spatial and temporal features of the solution behaves on the same order. Hence, one should have to incorporate anisotropic kernels. The nonlinear Burgers’ equations are converted by nonlinear transformation to linear equations. The spatial discretizations are carried out to construct differentiation matrices. Comparisons with most available numerical methods are made to solve the Burgers’ equations.

1. Introduction

Nonlinear partial differential equations can be approximated by various numerical methods such as finite volume methods, finite element methods, meshless methods, boundary element methods, wavelets methods, methods of fundamental solutions, and spectral methods (see [1,2,3,4,5,6,7,8,9] and references therein). In all these approaches, the time derivative is discretized using Runge–Kutta methods, Lie splitting methods, and various explicit and implicit methods. Some work is available in the literature to approximate space derivatives as well as time derivatives. For example, a space–time formulation was developed by Netuzhylov (see [10]); the author used the inverse moving least squares (IMLS) method to solve the problem of moving boundaries. Li and Mao [11] used RBFs on the space–time domain and developed a space–time global RBF method to solve inverse problems. Their resultant system was not square, so they used the least square method to overcome the ill-posedness of the problem. Li’s and Mao’s techniques were further used in the work [12] to solve other inverse problems. Furthermore, in [13], the authors developed a method to estimate river pollution using global space–time RBFs. Various types of space–time formulations can be found in [14,15,16,17,18]. Recently, the authors in the work [19,20] developed efficient space–time methods for solving time-dependent PDEs. In this work, a space–time numerical scheme is constructed using anisotropic kernels to solve nonlinear Burgers’ equations. The time-dependent PDEs are collocated using space–time kernels in both space and time dimensions. We tested our approach to solve 1-D and 2-D Burgers’ equations.

2. Space–Time Kernel-Based Approximation

The time-dependent PDE in the spatial domain ${\mathsf{\Omega }}_d\in {\mathrm{R}}^d$, where $(d\ge 1)$ and for time interval $(0,T_f)$ satisfies the equation

$$(\frac{\partial }{\partial t}-\mathcal{L})u(t,x)=f(t,x),(t,x)\in \mathsf{\Omega }=(0,T_f)\times {\mathsf{\Omega }}_d,$$

(1)

along with some well defined initial and boundary conditions. Here, $\mathcal{L}$ is some spatial operator. The time-dependent problem is seen as the problem in $d+1$-dimensions. The summarized form of Equation (1) along with the boundary and the initial condition is given by the following:

$${\mathcal{L}}_1\left[u\left(\widehat{x}\right)\right]=f\left(\widehat{x}\right),\widehat{x}=(t,\mathbf{x})\in \mathsf{\Omega },t>0$$

(2)

$$\mathcal{B}\left[u\left(\widehat{x}\right)\right]=g\left(\widehat{x}\right),\widehat{x}=(t,\mathbf{x})\in \partial \mathsf{\Omega },t>0$$

(3)

$$u\left(\widehat{x}\right)=u_0,\widehat{x}=(t,\mathbf{x})\in \mathsf{\Omega },t=0.$$

(4)

These give rise to a well-posed problem (since the resulting system is square) that is used to approximate the solution to $u\left(\widehat{x}\right)$ of our problem. The space–time formulation combines space and time variables so that the functional space on $\mathsf{\Omega }=[0,T]\times {\mathsf{\Omega }}_d\subset {\mathbf{R}}^{d+1}$ contains the unknown function $u\left(\widehat{x}\right)$. The following functional space is incorporated in our work.

$$W_h=\mathrm{span}\{\varphi (\parallel .-{\widehat{x}}_1^c\parallel ),\varphi (\parallel .-{\widehat{x}}_2^c\parallel ),\dots \varphi (\parallel .-{\widehat{x}}_N^c\parallel )\}$$

(5)

where $\varphi$ is radial kernel, and $\parallel .\parallel$ is any norm. We have used the Euclidean norm, and points in $\{{\mathbf{x}}_1^c,{\mathbf{x}}_2^c,\dots {\mathbf{x}}_N^c\}\subset \mathsf{\Omega }$ are centers. The solution $u\left(\widehat{x}\right)$ at the point $\widehat{x}\in \mathsf{\Omega }$ is then defined by

$$u\left(\widehat{x}\right)=\sum _{j=1}^N{a}_j\varphi (\parallel \widehat{x}-{\widehat{x}}_j^c\parallel ).$$

(6)

The space–time kernel-based method requires the approximated unknown to satisfy equations as well as initial and boundary conditions at all nodes in domain $\mathsf{\Omega }$. For simplicity, we take $N_I$ internal nodes, $N_B$ boundary nodes, and $N_T$ initial time $(t=0)$ nodes. The unknown $u\left(\widehat{x}\right)=u(t,\mathbf{x})$ is used to satisfy the equation by putting Equations (2) and (3). We then have

$${\mathcal{L}}_{1}u\left({\widehat{x}}_i\right)=\sum _{j=1}^N{a}_j{\mathcal{L}}_1\varphi (\parallel {\widehat{x}}_i-{\widehat{x}}_j^c\parallel )=f\left({\widehat{x}}_i\right),i=1,\dots ,N_I$$

(7)

$$\sum _{j=1}^N{a}_j\mathcal{B}\varphi (\parallel {\widehat{x}}_i-{\widehat{x}}_j^c\parallel )=g\left({\widehat{x}}_i\right),i=N_{I+1},\dots ,N_I+N_B$$

(8)

$$\sum _{j=1}^N{a}_j\varphi (\parallel {\widehat{x}}_i-{\widehat{x}}_j^c\parallel )=u_0,i=N_I+N_B+1,\dots ,N.$$

(9)

Equations (7)–(9) give rise to a linear algebraic system of equations with $N\times N$ equations and N unknowns. In a more compact form, we obtain

$$\left(\begin{array}{c}{M}_L\\ M_B\\ M_I\end{array}\right)\mathbf{a}=\left(\begin{array}{c}f\\ g\\ u_0\end{array}\right)$$

(10)

or

$$\mathbf{M}\mathbf{a}=\mathbf{b}.$$

(11)

The system of linear Equation (11) can be solved for unknown coefficient $\mathbf{a}$ using an equation solver. The solution to Equations (2)–(4) can be obtained from Equation (6).

3. Stability of the Numerical Scheme

To discuss the stability of our numerical scheme expressed in Equation (11), which is given by

$$\mathbf{M}\mathbf{a}=\mathbf{b}$$

(12)

where M is $N\times N$ differentiation matrix, the stability constant of Equation (12) is defined by

$$c_s=\underset{\mathbf{a}\ne 0}{sup}\frac{\parallel \mathbf{a}\parallel }{\parallel \mathbf{M}\mathbf{a}\parallel }.$$

(13)

The value $c_s$ is bounded for every discrete norms $\parallel .\parallel$ on ${\mathbb{R}}^N$. Hence, we have

$${\parallel \mathbf{M}\parallel }^{-1}\le \frac{\parallel \mathbf{a}\parallel }{\parallel \mathbf{M}\mathbf{a}\parallel }\le c_s.$$

(14)

Similarly, in the case of pseudoinverse ${\mathbf{M}}^{†}$ of $\mathbf{M}$, we obtain

$$\parallel {\mathbf{M}}^{†}\parallel =\underset{\mathbf{v}\ne 0}{sup}\frac{\parallel {\mathbf{M}}^{†}\mathbf{v}\parallel }{\parallel \mathbf{v}\parallel },$$

(15)

and we thus write

$$\parallel {\mathbf{M}}^{†}\parallel \ge \underset{\mathbf{v}=\mathbf{M}\mathbf{a}\ne 0}{sup}\frac{\parallel {\mathbf{M}}^{†}\mathbf{M}\mathbf{a}\parallel }{\parallel \mathbf{M}\mathbf{a}\parallel }=\underset{\mathbf{a}\ne 0}{sup}\frac{\parallel \mathbf{a}\parallel }{\parallel \mathbf{M}\mathbf{a}\parallel }=c_s.$$

(16)

Hence, Equations (14) and (16) provide the bounds for stability constant $c_s$.

4. Numerical Experiments

To validate the accuracy and robustness of our numerical scheme, we solve a nonlinear Burgers’ equation. The Hopf–Cole transformation [21] is used to transform a nonlinear Burgers’ equation into a heat equation. The solution accuracy greatly depends on the scale factor in the kernel function (for example, see [22,23] and references therein) The space–time method is applied with anisotropic radial kernels, namely anisotropic $C^2$ Matern. To achieve a desirable accuracy, we used two scale factors: one for the time variable and one for the space variable, as involved in the anisotropic kernels.

The anisotropic kernel $C^2$ Matern and its derivative, which has enough smoothness, are defined by

$$\varphi =(1+r)e^{-r},r=\sqrt{{\left({\epsilon }_{t}t\right)}^2+{\left({\epsilon }_{x}x\right)}^2},$$

$${\varphi }_t=-{\epsilon }_t^{2}t{e}^{-r},$$

$${\varphi }_x=-{\epsilon }_x^{2}x{e}^{-r},$$

$${\varphi }_{xx}=-{\epsilon }_x^2{e}^{-r}\left[-1+\frac{{\left(x{\epsilon }_x\right)}^2}{(r+eps)}\right]$$

where ${\epsilon }_x$ is the space shape parameter, ${\epsilon }_t$ is the temporal shape parameter, and $eps$ is a Matlab function. The accuracy of the space–time method is tested in terms of the maximum absolute error (MAE) defined by

$$\mathrm{MAE}={\mathrm{MAX}}_{1\le j\le N}\parallel u\left({\widehat{x}}_j\right)-\widehat{u}\left({\widehat{x}}_j\right)\parallel$$

(17)

where u and $\widehat{u}$ are the exact and the approximate solutions, respectively.

4.1. Example 1

We consider a Burgers’ equation defined by

$$w_t=\nu w_{xx}-w{w}_x,$$

(18)

with the following boundary conditions:

$$w(t,0)=w(t,1)=0,t>0$$

(19)

and the initial condition

$$w(0,x)=sin\left(\pi x\right),0<x<1.$$

(20)

Using the Hopf–Cole transformation [21,24],

$$w(t,x)=-2\nu \frac{u_x}{u}.$$

(21)

The nonlinear Equation (18) is converted into a linear heat equation

$$\frac{\partial u}{\partial t}=\nu \frac{{\partial }^{2}u}{\partial x^2},t>0,a<x<b,$$

(22)

with the boundary and initial conditions given by

$$u_x(t,0)=0,u_x(t,1)=0,t>0.$$

(23)

$$u(0,x)=exp\{-{\left(2\pi \nu \right)}^{-1}[1-cos\pi x)]\},0<x<1.$$

(24)

Hence, the solution u of the heat equation expressed in Equation (22) with conditions expressed in Equations (23) and (24) can be obtained using the space–time method discussed above. The solution to the Burgers’ equation can then be recovered from the transformation expressed in Equation (21). The exact solution to the linearized problem is defined by Equations (22)–(24) [25]:

$$u(t,x)={\alpha }_0+\sum _{j=1}^{\infty }{a}_{j}exp[-\left(j^2{\pi }^2\nu t\right)cos\left(j\pi x\right)]$$

(25)

where the coefficients are defined by

$${\alpha }_0={\int }_0^{1}exp\{-{\left(2\pi \nu \right)}^{-1}[1-cos\pi x)\}dx$$

(26)

and

$$a_j=2{\int }_0^{1}exp\{-{\left(2\pi \nu \right)}^{-1}[1-cos\pi x)]\}cos\left(j\pi x\right)dx,(j=1,2,3,\dots ).$$

(27)

Hence, using the Hopf–Cole transformation given by Equation (21), the (exact) Fourier solution to the problem given by Equations (18)–(20) is obtained as

$$w(t,x)=2\pi \nu \frac{{\sum }_{j=1}^{\infty }{\alpha }_{j}exp[-\left(j^2{\pi }^{2}vt\right)jsin\left(j\pi x\right)]}{{\alpha }_0+{\sum }_{j=1}^{\infty }{\alpha }_{j}exp[-\left(j^2{\pi }^2\nu t\right)cos\left(j\pi x\right)]}$$

(28)

where ${\alpha }_0$ and ${\alpha }_j(j=1,2,3,.\dots )$ are defined by Equations (26) and (27), respectively.

All results are obtained by the space–time meshless method using the anisotropic kernel with two scale factors: one in a time variable and one in a space variable. The results are displayed in Figure 1 and Figure 2 and

Table 1. The space–time method results and the results in [25] at $t=1$, $\nu =1$ in the space–time domain $[0,1]\times [0,1]$, where N denotes the number of collocation points, corresponding to Example 1.

		Space–Time Solution		Exact Solution	[25]
$\mathit{x}$	$\mathit{N}=\mathbf{25}$	$\mathit{N}=\mathbf{225}$	$\mathit{N}=\mathbf{400}$
0.1	0.1548	0.1105	0.109	0.1095	0.10954
0.2	0.2084	0.2113	0.2099	0.2098	0.20979
0.3	0.2915	0.2942	0.2925	0.2919	0.29190
0.4	0.3440	0.3502	0.3487	0.3479	0.34792
0.5	0.3753	0.3737	0.3723	0.3716	0.37158
0.6	0.3224	0.3608	0.3597	0.3591	0.35905
0.7	0.3022	0.3112	0.3104	0.3099	0.30991
0.8	0.2259	0.2283	0.2284	0.2278	0.22782
0.9	0.1261	0.1206	0.1215	0.1207	0.12069

, respectively. The results obtained by the space–time kernel-based method are comparable to the exact solution and the results in [25]. The main advantage of the current method is the avoidance of a time stepping procedure that requires a small time step for higher accuracy and stability. On the contrary, here the dimensions of our problem is increased by 1, yet it faces no difficulty, as these kernel-based methods are designed for multi-dimensional domains. The second advantage is that we can apply the method for irregular domains as well.

4.2. Example 2

As a second test problem, we considered the Burgers’ Equation in Equation (18) with the exact solution

$$w(t,x)=\frac{2vak{e}^{-v{k}^{2}t}coskx}{b+a{e}^{-v{k}^{2}t}coskx}.$$

(29)

The initial and boundary conditions can be extracted from the exact solution. The Hopf–Cole transformation converts the Burgers’ equation to the heat equation

$$\frac{\partial u}{\partial t}=\nu \frac{{\partial }^{2}u}{\partial x^2},$$

(30)

with initial condition

$$u(0,x)=b+acoskx.$$

(31)

The solution to the heat equation for the specified initial condition is given by

$$u(t,x)=b+a{e}^{-v{k}^{2}t}coskx$$

(32)

where $b>\left|a\right|$ to ensure that $u(t,x)>0$ for all time. The solution to the Burgers equation can now be easily obtained using the Hopf–Cole transformation

$$w(t,x)=-2\nu \frac{u_x}{u}=\frac{2\nu ak{e}^{-\nu k^{2}t}coskx}{b+a{e}^{-\nu k^{2}t}coskx}.$$

(33)

The space–time meshless method is used to solve the test problem given in Equation (2) incorporating anisotropic kernel with two scale factors: one in the time variable and one in the space variable, as shown in Figure 3 and Figure 4 and

Table 2. Numerical results when $t=1$, $a=1$, and $b=1$ in the space–time domain $[0,1]\times [0,1]$, where $N_x$ points in the $x-$ direction and $N_t$ in the $t-$ direction denotes the number of collocation points in the space–time domain, corresponding to Example 2.

${\mathit{N}}_{\mathit{x}}$	${\mathit{N}}_{\mathit{t}}$	N	${\mathbf{L}}_{\infty }$	${\mathbf{L}}_2$	C.time
5	10	50	3.83 $\times {10}^{-2}$	5.63 $\times {10}^{-2}$	0.1213
10	15	150	1.43 $\times {10}^{-2}$	2.93 $\times {10}^{-2}$	0.1757
20	30	600	6.30 $\times {10}^{-3}$	1.70 $\times {10}^{-2}$	2.3641
30	40	1200	4.97 $\times {10}^{-3}$	1.24 $\times {10}^{-3}$	6.2123
40	50	2000	1.50 $\times {10}^{-4}$	3.90 $\times {10}^{-4}$	13.545
50	60	3000	5.73 $\times {10}^{-4}$	3.41 $\times {10}^{-4}$	20.102

, respectively. It is observed that the space–time kernel-based method results are comparable to exact solutions.

4.3. Example 3

In the last test example, we consider a 2D Burgers’ equation that can be transformed into the 2D heat equation using Hopf–Cole transformation [26]. We consider the transformed 2D heat equation to validate the space–time kernel-based method for anisotropic kernels:

$$\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2},$$

(34)

with initial and boundary conditions defined by

$$u(0,x,y)=sin\left(\pi x\right)sin\left(\pi y\right),0<x<1,0<y<1$$

(35)

$$u(t,0,y)=0,0<t<1,0<y<1$$

(36)

$$u(t,1,y)=0,0<t<1,0<y<1$$

(37)

$$u(t,x,0)=0,0<t<1,0<x<1$$

(38)

$$u(t,x,1)=0,0<t<1,0<x<1,$$

(39)

with an exact solution given by

$$u(t,x,y)=sin\left(\pi x\right)sin\left(\pi y\right)e^{-2{\pi }^{2}t},0<x,y<1,0<t<1.$$

(40)

We applied the space–time method to solve a 2D heat equation in the space–time domain ${[0,1]}^2\times [0,1]$. The results are displayed in Figure 5 and Figure 6 and

Table 3. Numerical results corresponding to Example 3, when domain ${[0,1]}^3$.

N	L${}_{\infty }$	L2	C.time
125	8.82 $\times {10}^{-2}$	1.60 $\times {10}^{-1}$	0.002791
216	1.61 $\times {10}^{-2}$	4.49 $\times {10}^{-1}$	0.005528
729	1.76 $\times {10}^{-2}$	7.11 $\times {10}^{-1}$	0.070214
1000	1.63 $\times {10}^{-2}$	7.67 $\times {10}^{-1}$	0.15588
1728	1.48 $\times {10}^{-2}$	8.42 $\times {10}^{-1}$	0.48465
3375	1.24 $\times {10}^{-2}$	8.93 $\times {10}^{-2}$	2.0845

. Once again we obtained very accurate results with ease and stability. The present method can work in multi-dimensions in irregular domains as well.

4.4. Concluding Remarks

In the present work, we constructed a numerical scheme for Burgers equations. The scheme is based on space–time anisotropic kernels. The time variable is also considered as a space variable, and this increases the space dimension by 1. This space–time method avoids time-integration methods such as $\theta$-weighted schemes and implicit and explicit numerical schemes. Such methods require a very small time step for stability and higher accuracy. The present numerical scheme reduces the cost of computation by avoiding the need to recompute the matrix for each time level, as contrary to methods such as the RK4 method for evolutionary PDEs.