Numerical solution for two-dimensional partial differential equations using SM’s method

Sanaullah Mastoi; Abdul Hamid Ganie; Abdulkafi Mohammed Saeed; Umair Ali; Umair Ahmed Rajput; Wan Ainun Mior Othman

doi:10.1515/phys-2022-0015

Open Access Published by De Gruyter Open Access March 8, 2022

Numerical solution for two-dimensional partial differential equations using SM’s method

Sanaullah Mastoi , Abdul Hamid Ganie , Abdulkafi Mohammed Saeed , Umair Ali , Umair Ahmed Rajput and Wan Ainun Mior Othman

From the journal Open Physics

https://doi.org/10.1515/phys-2022-0015

Abstract

In this research paper, the authors aim to establish a novel algorithm in the finite difference method (FDM). The novel idea is proposed in the mesh generation process, the process to generate random grids. The FDM over a randomly generated grid enables fast convergence and improves the accuracy of the solution for a given problem; it also enhances the quality of precision by minimizing the error. The FDM involves uniform grids, which are commonly used in solving the partial differential equation (PDE) and the fractional partial differential equation. However, it requires a higher number of iterations to reach convergence. In addition, there is still no definite principle for the discretization of the model to generate the mesh. The newly proposed method, which is the SM method, employed randomly generated grids for mesh generation. This method is compared with the uniform grid method to check the validity and potential in minimizing the computational time and error. The comparative study is conducted for the first time by generating meshes of different cell sizes, i.e., 10 × 10 , 20 × 20 , 30 × 30 , 40 × 40 using MATLAB and ANSYS programs. The two-dimensional PDEs are solved over uniform and random grids. A significant reduction in the computational time is also noticed. Thus, this method is recommended to be used in solving the PDEs.

Keywords: SM’s method; partial differential equation; fractional partial differential equation; finite difference method; uniform grids; randomly generated grids

Abbreviations

2D: two-dimensional
CGM: conjugate gradient method
FDM: finite difference method
GS: Gauss–Siedel
PDE: partial differential equation
SM’s method: Sanaullah Mastoi’s method

1 Introduction

The numerical solution of the partial differential equation (PDE) is mostly solved by the finite difference method (FDM). The FDM is an approximate numerical method to find the approximate solutions for the problems arising in mathematical physics [1], engineering, and wide-ranging phenomenon, including transient, linear, nonlinear and steady state or nontransient cases [2,3,4]. This method is applied to various geometries, distinct boundary conditions and bodies composed of different materials [5]. FDM applications can easily solve the complex discretization and mathematical modeling computational codes [6]. Therefore, FDM has been considered to be the critical method for solving practical models [7]. Many numerical solution techniques [8] are now available to solve PDEs after high-performance[9] computing development with extensive storage capability [10,11,12,13,14]. However, the most desirable method for the solution of PDEs is the FDM [6,15].

The heat equation model is an elliptic type PDE [16,17]. The engineering model is commonly solved using the FDM on uniform grids [18]. However, very few studies have been reported that have applied random step sizes [5,11]. The heat equation theory was developed for modeling how heat diffuses through a given region [19,20,21]. The homogeneous heat equation plays a vital role in studying heat conduction and other diffusion processes [6,22].

PDEs can be used to solve various engineering, mathematical, and medical problems involving multiple unknown variables [23,24,25,26]. The numerical schemes with theoretical analysis focus on the convergence of numerical solution and it is observed that the behavior of randomness was found better during random walk [27,28,29]. The various numerical systems are only consistent with limited Fourier steps, spatial steps and the converging steps during the continuous time and solution. The simulation was repeated several times and it was observed that random meshes could reduce the iteration due to their random behavior in cell sizes [30,31].

The numerical solution of PDE using the FDM over uniform meshes has been extensively studied. However, randomly generated grids are also proposed [32,33,34]. To the best of the authors’ knowledge, no study is found in the open literature that discusses the validity and potential of randomly generated grids over uniform grids. The solution of fractional PDEs [35,36], Laplace transform, the inverse Laplace transform method through Bernoulli polynomials [36,37,38], PDE solutions compared with the domain decomposition method, the θ (theta) method and nonlinear local PDEs [7,39] have previously been carried out using the FDM [37,40]. This study can be extended to the conjugate gradient method (CGM) and GMRES method. The CGM is used for one sample at each iteration [41,42,43]. The idea can be proposed on the GMRES method and CGM. The GMRES method is minimizing a step norm of the residual vector over the Krylov subspace. The CGM is used for large-scale data analysis, and the study focuses on the iteration of one sample and the norm of residuals [44,45,46].

The geometric representation for the grids (mesh) on the larger domain is discretized in smaller grids. Mesh types such as moving mesh and coupled moving mesh are applied to quasi-linear PDEs for the solution of problems such as magneto-hydrodynamics and the free-surface viscoelastic flow [47,48,49,50,51,52]. The mesh generation process needs advancement in engineering models for accuracy and less computational cost [53]. Additionally, strand meshes have also been used to solve problems involving the flow of multiphase, viscous, and low-speed fluids. However, the mesh generation process is just a past practice. Typically, selecting a suitable mesh generation process for a particular problem becomes difficult, as no established principle is available and various studies have discussed the mesh sizes used for solving scientific problems [33,54–58].

The FDM relies on the discretization of function on grids or meshes. There is no principle for mesh generations. Usually, the FDM uses uniform or regular grids, but the SM method focuses on randomly generated grids. A special treatment in solving the SM (Sanaullah Mastoi) method compared to the FDM is the discretization process of regular grids to random grids. In this method, we use mathematical software programs like MATLAB, ANSYS, etc.

The implementation of the SM method is by analyzing the effect of samples of random meshes on the convergence of the numerical solution of the differential equation.

In this study, we investigated the potential of using randomly generated grids. Furthermore, a detailed comparison with the uniformly generated grids is also performed to check the method’s validity.

2 Numerical methodology

2.1 Heat conduction model

A simple demonstration of the 2-D heat condition model is shown in Figure 1 [59]. In this model, the left bar behaves as a heat source and the right bar as a heat sink. Due to the temperature gradient between them, the heat flows from the heat source to the heat sink. Considering the two opposite faces (each of cross-sectional area A), the heat source is at temperature 100°C and the heat sink is at temperature 25°C, while the heat covers the length (m, n) in t seconds during the thermal transition phase.

Figure 1

An engineering model of heat conduction.

2.2 Finite difference formulation using one-dimensional (1D) differential equation

Consider the one-dimensional (1D) steady-state heat conduction as in Figure 2(a) by internal heat generation using Eq. (1):

(1) ∂ 2 u ∂ x 2 ( x ) = − u ( x ) , − s < x < + s , ∀ s ∈ N .

At node g , we approximate the first derivatives at points g − 1 2 Δ x and g + 1 2 Δ x as a function of u ( x ) .

Figure 2

The numerical solution of one-dimensional steady state using the SM method applied on the (a) ODE, (b) PDE solution at n = 20 , and (c) solution at n = 41 . The numerical solution of the proposed mathematical model (one-dimensional steady-state heat conduction), the SM method, is applied using MATLAB operators to generate grids randomly.

Figures 2(b) and (c) show the numerical solution versus the exact solution [19]. The mathematical steady-state heat conduction model used the grids at N = 20 and N = 41 having (random) unequal subintervals, respectively,

(2) d u d x g − 1 2 , h = u ( g , h ) − u ( g − 1 , h ) Δ x ,

(3) d u d x g + 1 2 , h = u ( g + 1 , h ) − u ( g , h ) Δ x ,

d 2 u d x 2 g ≈ d u d x g + 1 2 − d u d x g − 1 2 Δ x ≈ u ( g + 1 ) − u ( g ) Δ x – u ( g ) – u ( g − 1 ) Δ x Δ x ≈ u ( g + 1 ) − 2 u ( g ) + u ( g − 1 ) Δ x 2 .

Now the finite-difference approximation of the heat conduction equations is

(4) u ( g + 1 , h ) − 2 u ( g , h ) + u ( g − 1 , h ) Δ x 2 + u ( x ) = 0 .

We use the random step size for the randomly generated grids to solve one-dimensional heat equations to get random realizations:

(5) u q + 1 ( g , h ) = u q ( g + 1 , h ) + u q ( g − 1 , j ) − 2 u ( g , h ) k 2 ,

where q is the iteration index and q + 1 for successive iterations.

3 Finite difference formulation using two-dimensional (2D) differential equation

The two-dimensional PDE (heat equation) with initial and boundary conditions are

∂ 2 u ∂ x 2 ( x , y ) + ∂ 2 u ∂ y 2 ( x , y ) = − f ( x , y ) , 0 ≤ x ≤ a , 0 ≤ y ≤ b .

And the given conditions are

(6) u = 0 , u ( 0 , y ) = a 1 , u ( 1 , y ) = a 2 , u ( x , 1 ) = b 2 and u ( x , 0 ) = b 1 ,

where f is a given function, u is the dependant variable over x and y coordinates, a 1 , a 2 , b 1 and b 2 are the left, right, top and bottom BCs, respectively. The physical phenomena such as the distributions of temperature at a , b = 25°C , while c , d = 100°C can be seen in Figure 3.

Figure 3

(a) Two-dimensional structural mesh (m and n are the nodal points). (b) A unit square two-dimensional PDE.

3.1 Discretization of the heat equation using the FDM

Eq. (6) is discretized with the help of FDM:

(7) u ( i − 1 , j ) + u ( i + 1 , j ) − 2 u ( i , j ) h 2 + u ( i , j − 1 ) + u ( i , j + 1 ) − 2 u ( i , j ) k 2 = − u ( i , j ) ,

m ≥ i ≥ 1 , n ≥ j ≥ 1 . with u ( i , j ) = 0 ∀ i , j ∈ ℕ , u ( i , 1 ) = a 1 , u ( i , n ) = a 2 , u ( 1 , j ) = b 1 , u ( m , j ) = b 2 .

Figure 4 specifies the domain, which is discretized accordingly as an FDM mesh generation.

$Figure 4 (a) Domain discretized with m = n , m × n m=n,\hspace{.25em}m\times n grids and (b) domain discretized with m ≠ n , m × n m\ne n,\text{ }m\times n grids.$

Figure 4

(a) Domain discretized with m = n , m × n grids and (b) domain discretized with m ≠ n , m × n grids.

3.2 Uniform grids

Meshes are generated with the help of MATLAB. Codes were employed individually on the interior nodes by the Gauss–Seidel (GS) iterative method explicitly, as shown in Eq. (6):

(8) u p + 1 ( i , j ) = h 2 ( u p ( i + 1 , j ) + u p ( i − 1 , j ) ) + k 2 ( u p ( i , j + 1 ) + u p ( i , j − 1 ) ) 2 ( h 2 + k 2 ) − ( hk ) 2 ,

where p is an iteration and p + 1 for successive iterations.

After solving Eq. (7), sufficient uniform numerical results were obtained with x as i th and y as j th in equal step size generated increments. The step size is shown in Table 1.

Table 1

Uniform grids

S. no.	Mesh sizes ( m × n )	Step sizes ( d x , d y ) = ( h , k )	Nodes	No. of cells	Avg cell size
1	10 × 10	0.100	125	10 2	0.100
2	20 × 20	0.050	442	20 2	0.050
3	30 × 30	0.033	962	30 2	0.033
4	40 × 40	0.025	1,682	40 2	0.025

As shown in Table 1, uniform grids are used to solve 2D PDEs, and sample realization is used to test random grids’ practicability over uniform grids. The sample realizations are shown in Figure 5(a)–(d). Then, the numerical solution was executed on each mesh GS iterative technique. This method is known as the Liebmann method or the method of successive displacement, which is an iterative method for the approximation of the system of linear equations. However, it can be applied on the system or matrix with nonzero components on the diagonals; convergence is always guaranteed if and only if the matrix is either strictly symmetric or diagonally dominant [32] and positive significantly.

$Figure 5 Uniformly generated grids: (a) 10 × 10 , 10\times 10, (b) 20 × 20 20\times 20 , (c) 30 × 30 30\times 30 and (d) 40 × 40 40\times 40 .$

Figure 5

Uniformly generated grids: (a) 10 × 10 , (b) 20 × 20 , (c) 30 × 30 and (d) 40 × 40 .

3.3 Random grids

The random grids (random samples) or randomly generated grids are generated with each having grid sizes of 10 × 10, 20 × 20, 30 × 30, 40 × 40, 60 × 60, 70 × 70, 80 × 80, 90 × 90 and 100 × 100. Grids are generated with the help of MATLAB, with the built-in “rand” function (specified mesh size). In each realization, various cell sizes are used, namely h and k, as shown in Table 2.

Table 2

Randomly generated grids and iteration wise (random meshes versus uniform meshes)

Mesh size	Cell size		Standard deviation (cell size)	No. of iterations
Mesh size	Maximum	Average	Standard deviation (cell size)	Random grids	Uniform grids
10 × 10	0.14137	0.01	0.01269	114	125
20 × 20	0.03398	0.00250	0.00239	382	390
30 × 30	0.01301	0.00111	0.00079	646	732
40 × 40	0.02386	0.00063	0.00092	987	1116

The numerical solution for uniform meshes employed by Eq. (6), the condition changed for random grids due to the sudden change of step size (h in x directions and k in y directions), is shown in Figure 6. Different approaches are possible in this discretization; as seen in Figure 6, we have various neighboring grids u i , j , if we check in the left, right, top and bottom we have h 1 , h 2 , k 1 and k 2 , respectively. There are three distinct possible methodologies: mean or average (avg), minimum and maximum grids size are used in the random step size. The numerical solutions are required to compute bordered grids on different grid standards, where grids h i , k j are selected in ( h i − 1 , h i + 1 ) and ( k j − 1 , k j + 1 ) , respectively. The grids size in h, the required h min , h max , h average , and k min , k max , k average are computed in h i ( h i − 1 , h i + 1 ) and k j ( k j − 1 and k j + 1 ) , respectively. Following the estimation of the stored grids among h ( maximum ) = maximum ( h i − 1 , h i + 1 ) and h average = average ( h i − 1 , h i + 1 ) = mean ( h i − 1 , h i + 1 ) and similar for k, k ( maximum ) = maximum ( k i − 1 , k i + 1 ) and k average = average ( k j − 1 , k j + 1 ) = mean ( k j − 1 , k j + 1 ) . The selection of h max and k max found a better choice for the convergence of the novel methods in the approximate solution. So, Eq. (3) will be subsequently generating random grids,

(9) u r + 1 ( i , j ) = ( h max 2 u r ( i + 1 , j ) − u ( i − 1 , j ) ) + k max 2 ( u r ( i , j + 1 ) − u r ( i , j − 1 ) ) 2 ( h max 2 + k max 2 ) − ( hk ) 2 .

$Figure 6 Randomly generated grids (i = first realization, ii = second realization): (a) 10 × 10 10\times 10 , (b) 20 × 20 20\times 20 , (c) 30 × 30 30\times 30 and (d) 40 × 40 40\times 40 .$

Figure 6

Randomly generated grids (i = first realization, ii = second realization): (a) 10 × 10 , (b) 20 × 20 , (c) 30 × 30 and (d) 40 × 40 .

4 Statistical significance (uniform vs random meshes)

The randomly generated grids reduced the number of iterations and computational time compared to uniform grids. Therefore, the relationship between uniform grids and randomly generated grids justifies the significance of randomly generated grids. However, the number of iterations is established in the eighth-degree polynomial regression equation, where the parameters lie within the 95% confidence interval test. The numerical results of the goodness of fit for the uniformly and randomly generated grids on each realization are presented in Figure 7(a)–(j). The findings conclude that randomly generated grids will converge with decreased computational cost and converging iteration.

Figure 7

Regression fit for each mesh (uniform vs random) (a–d) are first to fourth realization, respectively.

The regression fit for the random grids shows that the linear regression consists of finding the best-fitting straight line through the uniform meshes versus random grid points.

5 Mathematical solutions

5.1 Analytical solution of PDEs

Analytical solutions of 2D PDEs are obtained through the separation of variables. Figure 8 represents an exact solution of 2D heat equations.

Figure 8

Analytical solutions of 2D PDEs.

5.2 Numerical solution

5.2.1 Numerical solution profiles over uniform meshes

The numerical solution of two-dimensional PDEs applied over uniform grids is defined in Figure 9(a)–(j). The cell sizes are 10 × 10 , 20 × 20 , …, 100 × 100 . The numerical solution is based on the temperature, where the temperature scale determines local profile solutions of grid deviates from 25 to 100°C.

$Figure 9 Local solution profile on the uniform mesh of size (a) 10 × 10 10\times 10 , (b) 20 × 20 20\times 20 , (c) 30 × 30 30\times 30 and (d) 40 × 40 40\times 40 .$

Figure 9

Local solution profile on the uniform mesh of size (a) 10 × 10 , (b) 20 × 20 , (c) 30 × 30 and (d) 40 × 40 .

The heat (or thermal) energy of a body with uniform properties is presented in the local profile solutions, where the temperature varies from 25 to 100°C.

5.2.2 Numerical solution profiles over random meshes

The numerical method for solving 2D PDEs using the FDM over random meshes is described in Figure 10(a)–(d) (i = first realization and ii = second realization). In this solution, we presented the solution based on two realizations of randomly generated grids of size 10 × 10 , 20 × 20 , 30 × 30 and 40 × 40 , respectively. The profile solutions obtained by randomly generated grids are compared with solutions obtained by uniform grids. The numerical solution is based on heat and temperature, where the temperature scale as local profile solutions of grids deviates from 25 to 100°C. The smoothness of the heat map in the solution proves the consistency of the solution. It has been noticed that the mesh size is increasing as the smoothness increases. This is the benchmark to solve the numerical scheme through random meshes. To the best of our knowledge, we have solved and compared the profile solutions of uniformly generated grids with randomly generated grids, for the first time. This method will help assess the practicability of randomly generated grids in physical life models.

$Figure 10 Local solution profile on random mesh of size (a) 10 × 10 10\times 10 , (b) 20 × 20 20\times 20 , (c) 30 × 30 30\times 30 and (d) 40 × 40 40\times 40 (i = first realization; ii = second realization).$

Figure 10

Local solution profile on random mesh of size (a) 10 × 10 , (b) 20 × 20 , (c) 30 × 30 and (d) 40 × 40 (i = first realization; ii = second realization).

The local profile solutions on the random mesh size with random realizations in the heat equation deviate from the temperature.

5.3 Computational time and percentage deductions in computational time

The randomly generated grid over uniform grids is analyzed, and the key output parameters (i.e., converging iterations, computational time (s) and percentage (%) reduction in computational time) are compared in the numerical solution. The improving iterations and the computational time (uniform versus randomly generated grids) are presented in Table 3. The computational time for the cell sizes could reduce up to 43% from uniformly generated grids to randomly generated grids

Table 3

Computational time and number of iterations

S. no	Cell size	Converging iteration		Computational time (s)		Percentage reduction (%)
S. no	Cell size	Uniform grids	Random grids	Uniform grids	Random grids	Percentage reduction (%)
1	10 × 10	125	114	0.0189	0.0147	22.22
2	20 × 20	382	390	0.1246	0.1102	11.55
3	30 × 30	646	732	0.3154	0.2652	15.9
4	40 × 40	987	1,116	0.6094	0.4401	27.78

6 Conclusion

This research compares the numerical solutions of the heat conduction equation and computational convergence over uniform and random meshes. The numerical solution uses the FDM over both random grids or randomly generated grids and uniformly generated grids. It has been observed that the numerical solutions obtained through randomly generated grids have shown fast convergence compared to the solutions achieved by uniform meshes. Therefore, the idea of getting a numerical solution using the randomly generated finite difference meshes has been tested and found feasible and practicable. Furthermore, the computational time for the cell sizes could reduce up to 43%, which is the achievement from uniformly generated grids to randomly generated grids. The idea of SM’s method has been extended and implemented in fractional calculus.

This research can further be extended in different directions to explore the randomly generated grids’ practicability over other methods. The sensitivity of random mesh parameters can also be analyzed. This study can be expanded in the CGM to use randomly generated grids. This research area needs further investigation into computational time and pointwise comparison of the numerical solution with uniform vs random meshes because sufficient research gaps are available.

Acknowledgements

The authors thank QUEST Pakistan and the Institute of Mathematical Science, Faculty of Science, University of Malaya, Malaysia, for providing an environment and facilities to fulfill the research goal.

Funding information: The authors thank QUEST University Pakistan for providing funding Grant Number (QUEST)/NH/FDP(ECL)/-67/07-03-2014 under the project of the faculty development program.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-09-17

Revised: 2022-01-29

Accepted: 2022-02-08

Published Online: 2022-03-08

This work is licensed under the Creative Commons Attribution 4.0 International License.

Numerical solution for two-dimensional partial differential equations using SM’s method

Abstract

Abbreviations

1 Introduction

2 Numerical methodology

2.1 Heat conduction model

2.2 Finite difference formulation using one-dimensional (1D) differential equation

3 Finite difference formulation using two-dimensional (2D) differential equation

3.1 Discretization of the heat equation using the FDM

3.2 Uniform grids

3.3 Random grids

4 Statistical significance (uniform vs random meshes)

5 Mathematical solutions

5.1 Analytical solution of PDEs

5.2 Numerical solution

5.2.1 Numerical solution profiles over uniform meshes

5.2.2 Numerical solution profiles over random meshes

5.3 Computational time and percentage deductions in computational time

6 Conclusion

Acknowledgements

References

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