Numerical Solutions to Wave Propagation and Heat Transfer Non-Linear PDEs by Using a Meshless Method

Abstract

The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method (GFDM). In this paper, the fully non-linear Eikonal equation and the stationary heat transfer equation with variable thermal conductivity and source term are solved in 2D. The explicit formulae for derivatives are developed and applied to the equations in order to obtain the numerical schemes to be used. Moreover, the numerical values that approximate the functions for the considered domain are obtained. Numerous examples for both equations on irregular 2D domains are exposed to underline the effectiveness and practicality of the method.

1. Introduction

It is well known that many problems given by non-linear PDEs appear in many fields of science [1,2]. In this paper, due to their importance, the numerical resolution of two of them is proposed: the Eikonal and stationary heat transfer equations. The increasing difficulty to solve these equations, due to the non-linearity of them, makes not only the application of existing numerical methods necessary but also makes the importance of carrying out new numerical methods more suitable for certain equations or domains.

The Eikonal equation is used to describe the behavior of different systems related to control, propagation of waves or environmental phenomena. The Eikonal equation takes the following form

$${\left|\nabla U\left(\mathbf{x}\right)\right|}^2=f\left(\mathbf{x}\right),$$

(1)

for some function f. H. Zhao proposed a fast sweeping method for computing the numerical solution of the Eikonal equations on a rectangular grid [3], in [4], the authors used a boundary-only meshless method, and in [5], a differential quadrature method was employed. Moreover, a fast marching method is proposed in [6] and a fast semi-Lagrangian method in [7].

The stationary heat transfer equation with thermal conductivity coefficient depending on the coordinates of the domain points and source or sink term describes heat transfer processes when the thermal conductivity is not uniform in the medium and, in addition, there is a heat source. We consider the stationary non-linear heat transfer equation

$$\nabla ·\left(\sigma \right(\mathbf{x})\nabla U(\mathbf{x}\left)\right)=-q\left(\mathbf{x}\right).$$

(2)

In (1) and (2), $\mathbf{x}\in \Omega$ is an open bounded domain in ${\mathbb{R}}^d$, for $d=1,2$. In [8,9,10,11], the authors show several methods for the numerical evaluation of the solutions of the heat transfer equation. Furthermore, in [12], a localized boundary-domain integro-differential equation approach is followed, and a wide summary of numerical methods for solving the stationary heat transfer equation is given in [13].

To the best of our knowledge, a more detailed review of several meshless methods are given in [14]. As an example, in [15], the authors proposed the meshless singular boundary method for anomalous diffusions equations (of particular interest are Remarks 3.2 and 3.3).

The GFDM is a meshless method, as it solves the PDE at each of the points (nodes) where the domain is discretized. Since it does not need integration, it does not need meshing either. Moreover, the addition of nodes for a better approximation of the solution is easy. In irregular domains, it has a great advantage to be able to add nodes in the subdomains where they are most needed. Obtaining the explicit expressions of the different partial derivatives (explicit formulas) at each node is simple and automatic. Meshless methods have great advantages and, undoubtedly, also disadvantages. Their development and applications have been very important recently. In this paper, we make use of the meshless finite difference approach called Generalized Finite Difference Method (GFDM), which is spreading and is being applied to different fields due to the difficulty of directly solving the partial derivative and time derivative terms on irregular domains. The method was proposed by Lizska and Orkisz [16] and has received great attraction since the paper by Benito et al. [17]. The discretization of the spatial partial derivatives uses a very simple expression (depending only on the distribution of a few nodes, as we explain in the next section), so the treatment of non-linearities is straightforward. For these reasons. a wide range of applications have been recently found [18,19,20]

The paper is organized in the following way. We present some key aspects of the GFDM in Section 2, as well as the application of the Newton–Raphson method. Several computational examples are drawn, and the order of the convergence of the proposed discretization is found. Moreover, the effectiveness of the method is exhibited. Finally, some conclusions are extracted.

2. GFDM and Newton–Raphson Method

2.1. Explicit Finite Difference Formulae

To obtain the explicit formulae of the GFDM, let $\Omega$ be a domain in ${\mathbb{R}}^2$ and $M=x_1,x_2,\cdots x_N$ a discretization of such domain. For the ease of notation and, without loss of generality, let us consider a set of s different points of M, say $V=\{{\mathbf{x}}_{0_1},\cdots ,{\mathbf{x}}_{0_s}\}\subset M$ for a generic interior point of $M-V$ as $x_0$. Here, with the previous notation, ${\mathbf{x}}_0=(x_0,y_0)$ and ${\mathbf{x}}_{0_i}=(x_i,y_i)$, define ${\xi }_i=x_i-x_0,{\eta }_i=y_i-y_0$. Different methods, such as distance, quadrant or octant criteria, are usually used to form the stars, see, for instance [17]. By minimizing the residual error of the second-order Taylor series with respect to the spatial partial derivatives at each point of $x_0$, the explicit difference formulae [17,21] are found, for any regular enough function $u\left(\mathbf{x}\right)$:

$$\left\{\begin{array}{c}{\displaystyle {\displaystyle \frac{\partial u(x_0,y_0)}{\partial x}}=-{\lambda }_{10}{u}_0+\sum _{i=1}^s{\lambda }_{1i}{u}_i+\Theta ({\zeta }_i^2,{\eta }_i^2),with{\lambda }_{10}=\sum _{i=1}^s{\lambda }_{1i},}\hfill \\ \\ {\displaystyle {\displaystyle \frac{\partial u(x_0,y_0)}{\partial y}}=-{\lambda }_{20}{u}_0+\sum _{i=1}^s{\lambda }_{2i}{u}_i+\Theta ({\zeta }_i^2,{\eta }_i^2),with{\lambda }_{20}=\sum _{i=1}^s{\lambda }_{2i},}\hfill \\ \\ {\displaystyle {\displaystyle \frac{{\partial }^{2}u(x_0,y_0)}{\partial x^2}}=-{\lambda }_{30}{u}_0+\sum _{i=1}^s{\lambda }_{3i}{u}_i+\Theta ({\zeta }_i^2,{\eta }_i^2),with{\lambda }_{30}=\sum _{i=1}^s{\lambda }_{3i},}\hfill \\ \\ {\displaystyle {\displaystyle \frac{{\partial }^{2}u(x_0,y_0)}{\partial y^2}}=-{\lambda }_{40}{u}_0+\sum _{i=1}^s{\lambda }_{4i}{u}_i+\Theta ({\zeta }_i^2,{\eta }_i^2),with{\lambda }_{40}=\sum _{i=1}^s{\lambda }_{4i}.}\hfill \end{array}\right.$$

(3)

Here, $\Theta ({\zeta }_i^2,{\eta }_i^2)$ denotes the local truncation error of order 2, as proven in [17,21]. By substituting these in the Eikonal Equation (1) or the stationary heat transfer Equation (2), we obtain a set of non-linear equations as follows:

$$H_p\left[u_q\right]=0,p,q=0,\cdots ,N.$$

(4)

2.2. Newton–Raphson Method (N–R)

We use the Newton–Raphson method for solving system (4).

Let us consider the Jacobian matrix

$${\mathit{J}}_H={\displaystyle \frac{\partial H_i\left(\mathit{u}\right)}{\partial u_j}}.$$

(5)

The N–R method is based on the convergence of the following

$${\mathit{U}}^{k+1}={\mathit{U}}^k-{\mathit{J}}_H^{-1}\left({\mathit{U}}^k\right)H\left({\mathit{U}}^k\right),$$

(6)

where $U^k$ stands for the series of solutionsin the $k-th$ iteration. To define the end of the iterations we consider two different criteria:

• When a given tolerance is reached, in this paper ${10}^{-3}$: $∥{\mathit{U}}^{k+1}-{\mathit{U}}^k∥\le {10}^{-3}$.
• When 15 iterations are attained.

The initial data ${\mathit{U}}^0$ is interpolated in the inside of the domain from the Dirichlet condition.

3. Numerical Examples for Eikonal Equation

In order to study the PDE

$$\left\{\begin{array}{ccc}\hfill & {\left|\nabla U\left(\mathbf{x}\right)\right|}^2=f\left(\mathbf{x}\right),\hfill & \hfill \mathbf{x}\in \Omega \subset {\mathbb{R}}^2,\\ \hfill & U_{\Gamma }=g\left(\mathbf{x}\right),\hfill & \hfill \mathbf{x}\in \Gamma ,\end{array}\right.$$

(7)

we use as boundary condition the value of the known solution at those nodes, and we compute the numerical errors with the following

$$\left\{\begin{array}{c}{l}_2=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^N{\left(u_i-U_i\right)}^2}}{N}}},\hfill \\ l_{\infty }=max|u_i-U_i|,\hfill \end{array}\right.$$

(8)

where $u_i$ is the GFD solution and $U_i$ is the exact solution at node i.

3.1. Numerical Test 1

Consider the following PDE

$${\left({\displaystyle \frac{\partial U}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial U}{\partial y}}\right)}^2=y^2{e}^{-x}\left(4+y^2\right)$$

(9)

with the exact solution

$$U(x,y)=y^2{e}^{-x}.$$

(10)

We employ the meshes of Figure 1. The norms of the errors ar shown in

Table 1. Error norms for Example 1.

Cloud of Nodes	${\mathit{l}}_2$	${\mathit{l}}_{\mathit{\infty }}$	Iterations Number
Cloud 1	2.1354 × 10${}^{-5}$	5.2347 × 10${}^{-5}$	8
Cloud 2	5.3784 × 10${}^{-5}$	9.2317 × 10${}^{-5}$	6
Cloud 3	6.2513 × 10${}^{-5}$	1.0032 × 10${}^{-4}$	8
Cloud 4	1.2231 × 10${}^{-5}$	4.1256 × 10${}^{-5}$	7

.

3.2. Numerical Test 2

We solve the PDE

$${\left({\displaystyle \frac{\partial U}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial U}{\partial y}}\right)}^2={sin}^2\left(y\right)+x^2{cos}^2\left(y\right).$$

(11)

The exact solution is given by the explicit expression

$$U(x,y)=xsin\left(y\right).$$

(12)

The error norms are illustrated in

Table 2. Error norms for Example 2.

Cloud of Nodes	${\mathit{l}}_2$	${\mathit{l}}_{\mathit{\infty }}$	Iterations Number
Cloud 1	7.4231 × 10${}^{-5}$	2.5717 × 10${}^{-4}$	9
Cloud 2	9.3487 × 10${}^{-5}$	4.0196 × 10${}^{-4}$	8
Cloud 3	5.2066 × 10${}^{-5}$	8.9902 × 10${}^{-5}$	8
Cloud 4	1.2231 × 10${}^{-5}$	5.7086 × 10${}^{-5}$	7

.

4. Numerical Examples for Stationary Non-Linear Heat Transfer Equation

In this section, we present several numerical results obtained by solving the stationary non-linear heat transfer equation with a source term, where the conductivity is a function of the spatial coordinates, in 2D, with Dirichlet boundary conditions. In particular, we solve the PDE

$$\left\{\begin{array}{ccc}\hfill & \nabla ·\left(k\right(\mathbf{x})\nabla U)=-q\left(\mathbf{x}\right),\hfill & \hfill \mathbf{x}\in \Omega \subset {\mathbb{R}}^2,\\ \hfill & U_{\Gamma }=h\left(\mathbf{x}\right),\hfill & \hfill \mathbf{x}\in \Gamma ,\end{array}\right.$$

(13)

where $k\left(\mathbf{x}\right)$ is the thermal conductivity and $q\left(\mathbf{x}\right)$ is the source term.

4.1. Numerical Test 3

We consider the stationary non-linear heat transfer equation in the following form

$${\displaystyle \frac{\partial }{\partial x}}\left((x+y){\displaystyle \frac{\partial U}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left((x+y){\displaystyle \frac{\partial U}{\partial y}}\right)=e^{-x}\left(cosy-siny\right).$$

(14)

The analytical solution is

$$U(x,y)=e^{-x}siny.$$

(15)

In

Table 3. Error norms for Example 3.

Cloud of Nodes	${\mathit{l}}_2$	${\mathit{l}}_{\mathit{\infty }}$
Cloud 1	1.3154 × 10${}^{-5}$	3.4247 × 10${}^{-5}$
Cloud 2	3.33487 × 10${}^{-5}$	5.2914 × 10${}^{-5}$
Cloud 3	5.6244 × 10${}^{-5}$	8.3201 × 10${}^{-4}$
Cloud 4	8.1232 × 10${}^{-6}$	1.2572 × 10${}^{-5}$

, the error norms are plotted.

4.2. Numerical Test 4

We consider the stationary non-linear heat transfer equation in the following form:

$${\displaystyle \frac{\partial }{\partial x}}\left(xy{\displaystyle \frac{\partial U}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(xy{\displaystyle \frac{\partial U}{\partial y}}\right)=9xy(y-x)$$

(16)

which has the exact solution

$$U(x,y)=x^3-y^3.$$

(17)

Problem (16) is solved on the irregular clouds of Figure 1. In

Table 4. Error norms for Example 4.

Cloud of Nodes	${\mathit{l}}_2$	${\mathit{l}}_{\mathit{\infty }}$
Cloud 1	4.2578 × 10${}^{-5}$	6.7241 × 10${}^{-5}$
Cloud 2	7.1378 × 10${}^{-5}$	1.0058 × 10${}^{-4}$
Cloud 3	8.3254 × 10${}^{-6}$	1.9231 × 10${}^{-5}$
Cloud 4	1.9288 × 10${}^{-5}$	3.4255 × 10${}^{-5}$

the error norms are illustrated.

4.3. Convergence Test

In this subsection we provide computational examples showing the numerical order of convergence. We solve the equation

$${\displaystyle \frac{\partial }{\partial x}}\left((x^2+y^2){\displaystyle \frac{\partial U}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left((x^2+y^2){\displaystyle \frac{\partial U}{\partial y}}\right)=-{\displaystyle \frac{2x+2y+4xy}{{(1+x+y)}^2}}$$

(18)

with exact solution

$$U(x,y)=ln(1+x+y).$$

(19)

For our purposes, we generate two irregular clouds of points from the third cloud of Figure 1. In

Table 5. Error norms $l_2$ and $l_{\infty }$ for convergence test.

Number of Nodes	${\mathit{l}}_2$	${\mathit{l}}_{\mathit{\infty }}$
55	0.002973	0.007653
197	0.000810	0.002129
743	0.000206	0.000608

, we show the number of nodes, N, $l_2$ and $l_{\infty }$ norms for the three clouds of points in Figure 2, to which we have performed the refinement.

In Figure 3, we display the $l_2$ and $l_{\infty }$ norms as functions of the number of nodes N for the three clouds of nodes. The vertical axis of the figures has been plotted on a logarithmic scale.

5. Example 5: Physical Example

Finally, we proposed a physical example showing the applicability of the above method. To this end, consider the following stationary heat equation

$${\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+2x{\displaystyle \frac{{\partial }^{2}T}{\partial x}}=0,$$

where $T(x,y)$ is the temperature in the square $[0,1]\times [0,1]$. The boundary conditions are

$$\left\{\begin{array}{c}0,(x,y)\in [0,1]\times \{y=0\},\hfill \\ \\ 100,(x,y)\in [0,1]\times \{y=1\},\hfill \\ \\ 0,(x,y)\in \{x=0\}\times [0,1],\hfill \\ \\ 100,(x,y)\in \{x=1\}\times [0,1],\hfill \end{array}\right.$$

The solution of the physical example is plotted in Figure 4.

6. Conclusions

In order to solve the fully non-linear Eikonal and stationary heat transfer equations, the explicit formulas of the GFDM (Generalized Finite Difference Method) are applied using different irregular clouds of points.

The convenience of using the GFDM meshless method to solve the Eikonal and stationary heat transfer equations are shown by the efficiency and accuracy of the results provided by the application of the method. In fact, the accuracy in the results has been validated in different examples (different domains and solutions).

Apart from applying the method to different domains with irregular clouds, an additional example using the same domain but three different clouds of points (with different point density or refinement) has been studied in order to analyze the convergence of the proposed methodology.