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., 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.
conjugate gradient method
finite difference method
partial differential equation
- SM’s method
Sanaullah Mastoi’s method
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 , 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 . FDM applications can easily solve the complex discretization and mathematical modeling computational codes . Therefore, FDM has been considered to be the critical method for solving practical models . Many numerical solution techniques  are now available to solve PDEs after high-performance 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 . 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 (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 . 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 . 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.
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):
At node , we approximate the first derivatives at points and as a function of .
Figures 2(b) and (c) show the numerical solution versus the exact solution . The mathematical steady-state heat conduction model used the grids at and having (random) unequal subintervals, respectively,
Now the finite-difference approximation of the heat conduction equations is
We use the random step size for the randomly generated grids to solve one-dimensional heat equations to get random realizations:
where is the iteration index and 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
And the given conditions are
where is a given function, is the dependant variable over x and coordinates, and are the left, right, top and bottom BCs, respectively. The physical phenomena such as the distributions of temperature at , while can be seen in Figure 3.
3.1 Discretization of the heat equation using the FDM
Eq. (6) is discretized with the help of FDM:
Figure 4 specifies the domain, which is discretized accordingly as an FDM mesh generation.
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):
where is an iteration and for successive iterations.
After solving Eq. (7), sufficient uniform numerical results were obtained with as th and as th in equal step size generated increments. The step size is shown in Table 1.
|S. no.||Mesh sizes||Step sizes||Nodes||No. of cells||Avg cell size||Sd (standard deviation)|
|2||20 × 20||0.050||442||0.050||0|
|3||30 × 30||0.033||962||0.033||0|
|4||40 × 40||0.025||1,682||0.025||0|
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  and positive significantly.
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.
|Mesh size||Cell size||Standard deviation (cell size)||No. of iterations|
|Maximum||Average||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 , if we check in the left, right, top and bottom we have and , 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 are selected in and , respectively. The grids size in h, the required , , and , are computed in and , respectively. Following the estimation of the stored grids among and and similar for k, and . The selection of and found a better choice for the convergence of the novel methods in the approximate solution. So, Eq. (3) will be subsequently generating random grids,
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.
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.
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 , …, The numerical solution is based on the temperature, where the temperature scale determines local profile solutions of grid deviates from 25 to 100°C.
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 , , and , 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.
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
|S. no||Cell size||Converging iteration||Computational time (s)||Percentage reduction (%)|
|Uniform grids||Random grids||Uniform grids||Random grids|
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.
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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