In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the one-dimensional linear and nonlinear convection–diffusion problems, which frequently occur in physical phenomena. This is fulfilled by constructing a simple ‘anisotropic’ space–time Gaussian radial basis function. According to the proposed scheme, there is no need to remove time-dependent variables during the whole solution process, which leads it to a really meshless method. The suggested meshless method is implemented to the challenging convection–diffusion problems in a direct way with ease. Numerical results show that the proposed meshless method is simple, accurate, stable, easy-to-program and efficient for both linear and nonlinear convection–diffusion equation with different values of Péclet number. To assess the accuracy absolute error, average absolute error and root-mean-square error are used.
The convection–diffusion, advection–diffusion, or drift-diffusion equations have been playing a significant role in many engineering applications. The energy can be transformed inside a physical system due to the convection and diffusion processes to describe physical phenomena.
For a large variety of problems in every subjects, it is almost impossible to get the analytical solutions for changing in time and transport processes [23,24]. Numerical approximations are alternative to the analytical solutions for convection–diffusion equations. In literatures, there are several numerical techniques for solving the convection–diffusion equations [33,41]. These numerical techniques are based on the finite-difference approximations [25,32], integral transform methods [28,29], Monte Carlo simulation [4,5], variational iteration methods [6,7, 8,9,10], and meshless method using radial basis functions [11, 12,13,14, 15,16,17, 18,19,20].
The behavior of high-order time-stepping methods combined with mesh-free methods is studied for the transient convection–diffusion equation [31]. A Petrov–Galerkin method and Green’s functions are used to solve convection–diffusion problems [21]. A new approach to construct a stable RKPM method for convection-dominated problems is presented in ref. [30]. The space–time least-squares finite element methods are constructed for the advection–diffusion equation by using both linear shape functions and quadratic B-spline shape functions [22]. The particle transport method is developed for solving linear convection problems [44]. Several literatures focus on the investigation of different splines as interpolating function for solving one-dimensional advection–diffusion equations [34,37]. These literatures focus on the one-dimensional convection–diffusion–reaction equations with constant diffusion coefficient. Recently, a high-accuracy adaptive difference strategy is investigated by Zhu and Rui [47] on 1D convection–diffusion–reaction equation with convection item. It can explain the quenching phenomena of nonlinear singular degenerate problems. Pourgholi et al. [43] proposed a meshless method using radial basis functions method based on the finite-difference method to solve a nonlinear inverse convection–reaction–diffusion problem with an unknown source function. These numerical techniques are based on two-level finite difference approximations.
It is well-known that the radial basis function methods are attractive in numerical simulation due to their simple, flexible, and truly meshfree features. In this study, we propose a direct meshless method with one-level approximation, based on the radial basis functions, for the one-dimensional linear and nonlinear convection–diffusion problems. This is fulfilled by considering the time variable as a normal space variable. There is no need to remove the time-dependent variable during the whole solution process. Under this scheme, we can solve the convection–diffusion problems in a direct way.
The structure of this paper is organized as follows. Followed by Section 2, we introduce the formulation of the direct radial basis function (DMM) with space–time distance. Section 3 presents the methodology for convection–diffusion problems under initial conditions and boundary conditions. Section 4 examines several linear with different Péclet numbers and nonlinear problems. Several numerical examples are presented to validate the accuracy and stability of the proposed algorithm for one-dimensional linear and nonlinear convection–diffusion problems. Some conclusions are given in Section 5 with some additional remarks.
To describe the interaction between convection effects and diffusion transports, we can get the general mathematical formulation of convection–diffusion–reaction problem
For traditional numerical techniques, equation (1) should be discretized using the finite difference method or integral transform method, which leads to a steady-state equation. Then, the other numerical techniques can be used to get the numerical solutions. This provides a two-level procedure. To obtain a one-level procedure, we propose a direct collocation scheme by using the Gaussian radial basis function (GRBF).
For direct RBF-based collocation methods, the approximate solution can be written as a linear combination of RBFs for the approximation space under consideration for 2D or more higher-dimensional problems. We take the following GRBF for 2D problems as an example ref. [27]
However, there is only one space variable
Configuration of the space–time coordinate.
In the literature, there is a product model of a space–time radial basis function [42], which was introduced by Myers et al. [39,40],
The other types of definitions of radial or non radial space–time radial basis functions can be found in ref. [35,36].
Based on the definition of space–time radial basis functions, the above-mentioned equations (1)–(3) can be solved directly in a one-level approximation. Thus, the approximate solution of the function
To compare with the previous literatures, we consider using the absolute error, root-mean-square error (RMSE) [45,46], and average absolute error (AAE) defined as follows:
In the first two examples, we consider two linear cases with governing equation
This example considers the following initial condition
The corresponding analytical solution is given by
For fair comparison, the dimensionless Péclet number is defined as
Numerical results with
t |
|
|
|
|
|
AAE | |
---|---|---|---|---|---|---|---|
0.1 | TPS | 0.0834 | 0.0451 | 0.0244 | 0.0132 | 0.0069 | 0.0000 |
FD | 0.0836 | 0.0454 | 0.0244 | 0.0133 | 0.0073 | 0.0001 | |
CS | 0.0848 | 0.0452 | 0.0244 | 0.0132 | 0.0077 | 0.0003 | |
DMM | 0.0833 | 0.0451 | 0.0244 | 0.0132 | 0.0072 |
|
|
0.5 | TPS | 0.1376 | 0.0744 | 0.0402 | 0.0216 | 0.0113 | 0.0001 |
FD | 0.1384 | 0.0757 | 0.0413 | 0.0225 | 0.0123 | 0.0008 | |
CS | 0.1407 | 0.0757 | 0.0407 | 0.0223 | 0.0131 | 0.0010 | |
DMM | 0.1374 | 0.0744 | 0.0402 | 0.0218 | 0.0118 |
|
|
1 | TPS | 0.2570 | 0.1390 | 0.0751 | 0.0404 | 0.0211 | 0.0002 |
FD | 0.2587 | 0.1420 | 0.0778 | 0.0425 | 0.0232 | 0.0018 | |
CS | 0.2631 | 0.1420 | 0.0768 | 0.0421 | 0.0246 | 0.0023 | |
DMM | 0.2568 | 0.1389 | 0.0752 | 0.0407 | 0.0220 |
|
Numerical results with
t |
|
|
|
|
|
AAE | |
---|---|---|---|---|---|---|---|
0.1 | TPS | 0.1001 | 0.0782 | 0.0611 | 0.0478 | 0.0373 | 0.0000 |
FD | 0.1006 | 0.0788 | 0.0616 | 0.0481 | 0.0469 | 0.0013 | |
CS | 0.1005 | 0.0782 | 0.0611 | 0.0477 | 0.0372 | 0.0001 | |
DMM | 0.1000 | 0.0779 | 0.0607 | 0.0473 | 0.0368 |
|
|
0.5 | TPS | 0.1650 | 0.1289 | 0.1007 | 0.0787 | 0.0615 | 0.0001 |
FD | 0.1663 | 0.1316 | 0.1038 | 0.0817 | 0.0858 | 0.0042 | |
CS | 0.1661 | 0.1297 | 0.1011 | 0.0787 | 0.0613 | 0.0004 | |
DMM | 0.1649 | 0.1285 | 0.1001 | 0.0780 | 0.0607 |
|
|
1 | TPS | 0.3083 | 0.2409 | 0.1882 | 0.1470 | 0.1149 | 0.0001 |
FD | 0.3106 | 0.2460 | 0.1948 | 0.1545 | 0.1626 | 0.0087 | |
CS | 0.3102 | 0.2424 | 0.1893 | 0.1478 | 0.1152 | 0.0009 | |
DMM | 0.3081 | 0.2100 | 0.1870 | 0.1457 | 0.1135 |
|
As is known to all, when the Péclet number is low, the diffusion term dominates. In order to compare the DMM with the cubic
Average errors for DMM with different Péclet numbers
Pe |
|
|
|
|
|
---|---|---|---|---|---|
FD | 1/16 |
|
|
|
|
1/32 |
|
|
|
|
|
1/64 |
|
|
|
|
|
1/128 |
|
|
|
|
|
DMM | 1/16 |
|
|
|
|
Once the Péclet number increases, i.e., the convection term completely dominates over the diffusion term. In order to investigate problems with the far higher Péclet numbers, we compare the DMM results with the compact finite-difference approximation of fourth-order and the cubic
We consider the convection–diffusion equation (1) with the following initial condition:
The analytical solution is given by
Numerical results with different Péclet numbers
Pe | DMM | FD | DMM | FD | DMM | FD |
---|---|---|---|---|---|---|
x | 1,000 | 1,000 | 10,000 | 10,000 | 20,000 | 20,000 |
0.25 |
|
|
|
|
|
|
0.50 |
|
|
|
|
|
|
0.75 |
|
|
|
|
|
|
1.00 |
|
|
0 |
|
|
|
1.25 |
|
|
|
|
|
|
1.50 |
|
|
|
|
|
|
1.75 |
|
|
|
|
|
|
In this example, we consider a typical nonlinear convection–diffusion equation
We note that for smaller values of
Numerical results for
|
DMM | Uniform | Non-uniform |
---|---|---|---|
8 |
|
— | — |
16 |
|
— | — |
32 |
|
— | — |
64 |
|
|
|
128 | — |
|
|
256 | — |
|
|
For
In this study, a new direct meshless scheme is proposed for the one-dimensional linear and nonlinear convection–diffusion problems. The present numerical procedure, in which the time variable is considered as normal space variable, is based on the Gaussian radial basis function. There is no need to remove time-dependent variable during the whole solution process. Numerical results for several typical examples show that the proposed method is better than some other numerical methods given in the recent literature in terms of solution accuracy, stability and efficiency for the linear convection–diffusion equation with different values of Péclet number. These results lead us that the proposed method can successfully be used to nonlinear problems with accurate numerical results.
This work was supported by the Natural Science Foundation of Anhui Province (Project No. 1908085QA09) and the University Natural Science Research Project of Anhui Province (Project No. KJ2019A0591 & KJ2020B06).
Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.
Data availability statement: The data used to support the findings of this study are available from the corresponding author upon request.
[1] Boztosun I , Charafi A , Boztosun D . On the numerical solution of linear advection–diffusion equation using compactly supported radial basis functions. In: Griebel M , Schweitzer MA (eds) Meshfree methods for partial differential equations. Lecture notesin computational science and engineering, vol 26. Berlin, Heidelberg: Springer; 2003. Search in Google Scholar
[2] Boztosun I , Charafi A , Zerroukat M , Djidjeli K . Thin-plate spline radial basis function scheme for advection–diffusion problems. Electron J Bound Elements. 2002;BETEQ 2001:267–82. Search in Google Scholar
[3] Chen W , Hong YX , Lin J . The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method. Comput Math Appl. 2018;75:2942–54. Search in Google Scholar
[4] Fulger D , Scalas E , Germano G . Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space–time fractional diffusion equation. Phys Rev E. 2008;77(2):021122. Search in Google Scholar
[5] Koley U , Risebro NH , Schwab C , Weber F . A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations. J Hyperbolic Differ Equ. 2017;14(3):415–54. Search in Google Scholar
[6] Ahmad H , Khan TA , Ahmad I , Stanimirović PS , Chu Y-M . A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations. Results Phys. 2020;103462. 10.1016/j.rinp.2020.103462. Search in Google Scholar
[7] Ahmad H , Akgül A , Khan TA , Stanimirović PS , Chu Y-M . New perspective on the conventional solutions of the nonlinear time-fractional partial differential equations. Complexity. 2020;2020:8829017. 10.1155/2020/8829017. Search in Google Scholar
[8] Ahmad H , Khan TA , Stanimirović PS , Chu Y-M , Ahmad I . Modified variational iteration algorithm-II: convergence and applications to diffusion models. Complexity. 2020;2020:8841718. 10.1155/2020/8841718. Search in Google Scholar
[9] Ahmad H , Khan TA , Stanimirovic PS , Ahmad I Modified variational iteration technique for the numerical solution of fifth order KdV type equations. J Appl Comput Mech. 2020;6(SI):1220–7. Search in Google Scholar
[10] Ahmad H , Seadawy AR , Khan TA , Thounthong P . Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. J Taibah Univ Sci. 2020;14(1):346–58. Search in Google Scholar
[11] Ahmad I , Ahmad H , Inc M , Yao S-W , Almohsen B . Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer. Thermal Sci. 2020;24:95–105. 10.2298/TSCI20S1095A. Search in Google Scholar
[12] Shakeel M , Hussain I , Ahmad H , Ahmad I , Thounthong P , Zhang Y-F . Meshless technique for the solution of time-fractional partial differential equations having real-world applications. J Funct Spaces. 2020;8898309. 10.1155/2020/8898309. Search in Google Scholar
[13] Li JF , Ahmad I , Ahmad H , Shah D , Chu YM , Thounthong P , Ayaz M . Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method. Open Phys. 2020;18(1):1063–72. Search in Google Scholar
[14] Khan MN , Ahmad I , Akgül A , Ahmad H , Thounthong P . Numerical solution of time-fractional coupled Korteweg-de Vries and Klein-Gordon equations by local meshless method. Pramana-J Phys. 2021;95(1):1–13. Search in Google Scholar
[15] Ahmad I , Siraj-ul-Islam M , Zaman S . Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete & Continuous Dynamical Systems-S. 2020;13(10):2641. Search in Google Scholar
[16] Inc M , Khan MN , Ahmad I , Yao SW , Ahmad H , Thounthong P . Analysing time-fractional exotic options via efficient local meshless method. Results Phys. 2020;19:103385. 10.1016/j.rinp.2020.103385. Search in Google Scholar
[17] Ahmad I , Ahmad H , Thounthong P , Chu YM , Cesarano C . Solution of multi-term time-fractional PDE models arising in mathematical biology and physics by local meshless method. Symmetry. 2020;12(7):1195. Search in Google Scholar
[18] Ahmad I , Khan MN , Inc M , Ahmad H , Nisar KS . Numerical simulation of simulate an anomalous solute transport model via local meshless method. Alex Eng J. 2020;59(4):2827–38 Search in Google Scholar
[19] Khan MN , Ahmad I , Ahmad H . A radial basis function collocation method for space-dependent inverse heat problems. J Appl Comput Mech. 2020;6 (SI):1187–99. Search in Google Scholar
[20] Srivastava MH , Ahmad H , Ahmad I , Thounthong P , Khan NM . Numerical simulation of three-dimensional fractional-order convection–diffusion PDEs by a local meshless method. Therm Sci. 2020;210. 10.2298/TSCI200225210S Search in Google Scholar
[21] Cyron CJ , Nissen K , Gravemeier V , Wolfgang AW . Stable meshfree methods in fluid mechanics based on Greenas functions. Comput Mech. 2010;46:287–300. Search in Google Scholar
[22] Dag I , Irk D , Tombul M . Least-squares finite element method for the advection–diffusion equation. Appl Math Comput. 2006;173:554–65. Search in Google Scholar
[23] Wang FZ , Hou ER . A direct meshless method for solving two-dimensional second-order hyperbolic telegraph equations. J Math. 2020;2020:8832197. Search in Google Scholar
[24] Dehghan M . Numerical solution of the three-dimensional advection–diffusion equation. Appl Math Comput. 2004;150:5–19. Search in Google Scholar
[25] Ding HF , Zhang YX . A new difference scheme with high accuracy and absolute stability for solving convection–diffusion equations. J Comput Appl Math. 2009;230:600–6. Search in Google Scholar
[26] Fasshauer GE , Zhang JG . On choosing optimal shape parameters for RBF approximation. Numer Algorithms. 2007;45:345–68. Search in Google Scholar
[27] Fornberg B , Larsson E , Flyer N . Stable computations with Gaussian radial basis functions. SIAM J Sci Comput. 2011;33:869–92. Search in Google Scholar
[28] Glushkov EV , Glushkova NV , Chen CS . Semi-analytical solution to heat transfer problems using fourier transform technique, radial basis functions, and the method of fundamental solutions. Numer Heat Tr B. 2007;52:409–27. Search in Google Scholar
[29] Gu Y , He X , Chen W , Zhang C . Analysis of three-dimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method. Comput Math Appl 2018;75:33–44. Search in Google Scholar
[30] Hillman M , Chen JS . An implicit gradient meshfree formulation for convection-dominated problems. In: Bazilevs Y , Takizawa K (eds), Advances in computational fluid-structure interaction and flow simulation. Birkhäuser, Cham: Modeling and Simulation in Science, Engineering and Technology; 2016. Search in Google Scholar
[31] Huerta A , Fernandez-Mendez S . Time accurate consistently stabilized mesh-free methods for convection dominated problems. Int J Numer Methods Eng. 2001;50:1–18. Search in Google Scholar
[32] Karahan H . A third-order upwind scheme for the advection diffusion equation using spreadsheets. Adv Eng Softw. 2007;38:688–97. Search in Google Scholar
[33] Karahan H . Unconditional stable explicit finite difference technique for the advection–diffusion equation using spreadsheets. Adv Eng Softw. 2007;8:80–6. Search in Google Scholar
[34] Korkmaz A , Dag I . Cubic B-spline differential quadrature methods for the advection–diffusion equation. Int J Numer Methods H. 2012;22:1021–36. Search in Google Scholar
[35] Ku CY , Liu CY , Xiao JE , Chen MR . Solving backward heat conduction problems using a novel space–time radial polynomial basis function collocation method. Appl Sci. 2020;10:3215. Search in Google Scholar
[36] Liu CY , Ku CT , Xiao JE , Yeih WC . A novel spacetime collocation meshless method for solving two-dimensional backward heat conduction problems. CMES-Comp Model Eng. 2019;118:229–52. Search in Google Scholar
[37] Mohammadi R . Exponential B-spline solution of convection–diffusion equations. Appl Math. 2013;4:933–44. Search in Google Scholar
[38] Mohebbi A , Dehghan M . High-order compact solution of the one-dimensional heat and advection–diffusion equations. Appl Math Model. 2010;34:3071–84. Search in Google Scholar
[39] Myers DE . Anisotropic radial basis functions. Int J Pure Appl Math. 2008;42:197–203. Search in Google Scholar
[40] Myers DE , Iaco SD , Posa D , Cesare LD . Space-time radial basis functions. Comput Math Appl. 2002;43:539–49. Search in Google Scholar
[41] Nazir T , Abbas M , Ismail AIM , Majid AA , Rashid A . The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach. Appl Math Model. 2016;40:4586–611. Search in Google Scholar
[42] Parand K , Rad JA . Kansa method for the solution of a parabolic equation with an unknown space wise-dependent coefficient subject to an extra measurement. Comput Phys Commun. 2013;184:582–95. Search in Google Scholar
[43] Pourgholi R , Saeedi A , Hosseini A . Determination of nonlinear source term in an inverse convection–reaction–diffusion problem using radial basis functions method. Iran J Sci Technol A. 2019;43:2239–52. Search in Google Scholar
[44] Smolianski A , Shipilova O and Haario H . A fast high-resolution algorithm for linear convection problems: particle transport method. Int J Numer Methods Eng. 2007;70:655–84. Search in Google Scholar
[45] Wang FZ , Chen W , Jiang XR . Investigation of regularization techniques for boundary knot method. Commun Numer Meth En. 2010;26:1868–77. Search in Google Scholar
[46] Wang FZ , Chen W , Ling L . Combinations of the method of fundamental solutions for general inverse source identification problems. Appl Math Comput. 2012;219:1173–82. Search in Google Scholar
[47] Zhu XL , Rui HX . High-order compact difference scheme of 1D nonlinear degenerate convection–reaction–diffusion equation with adaptive algorithm. Numer Heat Tr B-Fund. 2019;75:43–66. Search in Google Scholar
© 2021 Fuzhang Wang et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.