Abstract
In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the onedimensional 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 timedependent 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, easytoprogram 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 rootmeansquare error are used.
1 Introduction
The convection–diffusion, advection–diffusion, or driftdiffusion 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 finitedifference 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 highorder timestepping methods combined with meshfree 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 convectiondominated problems is presented in ref. [30]. The space–time leastsquares finite element methods are constructed for the advection–diffusion equation by using both linear shape functions and quadratic Bspline 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 onedimensional advection–diffusion equations [34,37]. These literatures focus on the onedimensional convection–diffusion–reaction equations with constant diffusion coefficient. Recently, a highaccuracy 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 finitedifference method to solve a nonlinear inverse convection–reaction–diffusion problem with an unknown source function. These numerical techniques are based on twolevel finite difference approximations.
It is wellknown 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 onelevel approximation, based on the radial basis functions, for the onedimensional 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 timedependent 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 onedimensional linear and nonlinear convection–diffusion problems. Some conclusions are given in Section 5 with some additional remarks.
2 Formulation of the direct radial basis function
To describe the interaction between convection effects and diffusion transports, we can get the general mathematical formulation of convection–diffusion–reaction problem
where
and boundary conditions
For traditional numerical techniques, equation (1) should be discretized using the finite difference method or integral transform method, which leads to a steadystate equation. Then, the other numerical techniques can be used to get the numerical solutions. This provides a twolevel procedure. To obtain a onelevel procedure, we propose a direct collocation scheme by using the Gaussian radial basis function (GRBF).
For direct RBFbased collocation methods, the approximate solution can be written as a linear combination of RBFs for the approximation space under consideration for 2D or more higherdimensional problems. We take the following GRBF for 2D problems as an example ref. [27]
where
However, there is only one space variable
This can be easily extended to twodimensional or highdimensional cases.
Figure 1
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].
3 Methodology for DMM
Based on the definition of space–time radial basis functions, the abovementioned equations (1)–(3) can be solved directly in a onelevel approximation. Thus, the approximate solution of the function
To seek for the unknown coefficients
with
where
is
is
4 Numerical experiments
To compare with the previous literatures, we consider using the absolute error, rootmeansquare error (RMSE) [45,46], and average absolute error (AAE) defined as follows:
where
In the first two examples, we consider two linear cases with governing equation
4.1 Linear example 1
This example considers the following initial condition
and boundary conditions
The corresponding analytical solution is given by
and
For fair comparison, the dimensionless Péclet number is defined as
Table 1
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 

Table 2
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
Table 3
Pe 






FD  1/16 




1/32 





1/64 





1/128 





DMM  1/16 




4.2 Linear example 2
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 finitedifference approximation of fourthorder and the cubic
We consider the convection–diffusion equation (1) with the following initial condition:
and boundary conditions
The analytical solution is given by
For a fair comparison, we use the definition of dimensionless Péclet number
Table 4
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 






4.3 Nonlinear example 3
In this example, we consider a typical nonlinear convection–diffusion equation
with the exact solution
to measure the performance of the DMM by comparing with the method in ref. [47]. The source term
We note that for smaller values of
Table 5

DMM  Uniform  Nonuniform 

8 

—  — 
16 

—  — 
32 

—  — 
64 



128  — 


256  — 


For
5 Conclusions
In this study, a new direct meshless scheme is proposed for the onedimensional 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 timedependent 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.
Acknowledgments
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.
References
[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. 10.1007/9783642561030_5Search in Google Scholar
[2] Boztosun I , Charafi A , Zerroukat M , Djidjeli K . Thinplate 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. 10.1016/j.camwa.2018.01.023Search in Google Scholar
[4] Fulger D , Scalas E , Germano G . Monte Carlo simulation of uncoupled continuoustime random walks yielding a stochastic solution of the space–time fractional diffusion equation. Phys Rev E. 2008;77(2):021122. 10.1103/PhysRevE.77.021122Search in Google Scholar PubMed
[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. 10.1142/S021989161750014XSearch in Google Scholar
[6] Ahmad H , Khan TA , Ahmad I , Stanimirović PS , Chu YM . A new analyzing technique for nonlinear time fractional Cauchy reactiondiffusion 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 YM . New perspective on the conventional solutions of the nonlinear timefractional partial differential equations. Complexity. 2020;2020:8829017. 10.1155/2020/8829017. Search in Google Scholar
[8] Ahmad H , Khan TA , Stanimirović PS , Chu YM , Ahmad I . Modified variational iteration algorithmII: 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. 10.1080/16583655.2020.1741943Search in Google Scholar
[11] Ahmad I , Ahmad H , Inc M , Yao SW , Almohsen B . Application of local meshless method for the solution of two term time fractionalorder multidimensional 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 YF . Meshless technique for the solution of timefractional partial differential equations having realworld 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 twoterm timefractional PDE models arising in mathematical physics using local meshless method. Open Phys. 2020;18(1):1063–72. 10.1515/phys20200222Search in Google Scholar
[14] Khan MN , Ahmad I , Akgül A , Ahmad H , Thounthong P . Numerical solution of timefractional coupled Kortewegde Vries and KleinGordon equations by local meshless method. PramanaJ Phys. 2021;95(1):1–13. 10.1007/s12043020020255Search in Google Scholar
[15] Ahmad I , SirajulIslam M , Zaman S . Local meshless differential quadrature collocation method for timefractional PDEs. Discrete & Continuous Dynamical SystemsS. 2020;13(10):2641. 10.3934/dcdss.2020223Search in Google Scholar
[16] Inc M , Khan MN , Ahmad I , Yao SW , Ahmad H , Thounthong P . Analysing timefractional 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 multiterm timefractional PDE models arising in mathematical biology and physics by local meshless method. Symmetry. 2020;12(7):1195. 10.3390/sym12071195Search 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 10.1016/j.aej.2020.06.029Search in Google Scholar
[19] Khan MN , Ahmad I , Ahmad H . A radial basis function collocation method for spacedependent 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 threedimensional fractionalorder 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. 10.1007/s0046600904054Search in Google Scholar
[22] Dag I , Irk D , Tombul M . Leastsquares finite element method for the advection–diffusion equation. Appl Math Comput. 2006;173:554–65. 10.1016/j.amc.2005.04.054Search in Google Scholar
[23] Wang FZ , Hou ER . A direct meshless method for solving twodimensional secondorder hyperbolic telegraph equations. J Math. 2020;2020:8832197. 10.1155/2020/8832197Search in Google Scholar
[24] Dehghan M . Numerical solution of the threedimensional advection–diffusion equation. Appl Math Comput. 2004;150:5–19. 10.1016/S00963003(03)001930Search 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. 10.1016/j.cam.2008.12.015Search in Google Scholar
[26] Fasshauer GE , Zhang JG . On choosing optimal shape parameters for RBF approximation. Numer Algorithms. 2007;45:345–68. 10.1007/s1107500790728Search 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. 10.1137/09076756XSearch in Google Scholar
[28] Glushkov EV , Glushkova NV , Chen CS . Semianalytical 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. 10.1080/10407790701443859Search in Google Scholar
[29] Gu Y , He X , Chen W , Zhang C . Analysis of threedimensional anisotropic heat conduction problems on thin domains using an advanced boundary element method. Comput Math Appl 2018;75:33–44. 10.1016/j.camwa.2017.08.030Search in Google Scholar
[30] Hillman M , Chen JS . An implicit gradient meshfree formulation for convectiondominated problems. In: Bazilevs Y , Takizawa K (eds), Advances in computational fluidstructure interaction and flow simulation. Birkhäuser, Cham: Modeling and Simulation in Science, Engineering and Technology; 2016. 10.1007/9783319408279_3Search in Google Scholar
[31] Huerta A , FernandezMendez S . Time accurate consistently stabilized meshfree methods for convection dominated problems. Int J Numer Methods Eng. 2001;50:1–18. 10.1002/nme.602Search in Google Scholar
[32] Karahan H . A thirdorder upwind scheme for the advection diffusion equation using spreadsheets. Adv Eng Softw. 2007;38:688–97. 10.1016/j.advengsoft.2006.10.006Search 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. 10.1016/j.advengsoft.2006.08.001Search in Google Scholar
[34] Korkmaz A , Dag I . Cubic Bspline differential quadrature methods for the advection–diffusion equation. Int J Numer Methods H. 2012;22:1021–36. 10.1108/09615531211271844Search 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. 10.3390/app10093215Search in Google Scholar
[36] Liu CY , Ku CT , Xiao JE , Yeih WC . A novel spacetime collocation meshless method for solving twodimensional backward heat conduction problems. CMESComp Model Eng. 2019;118:229–52. 10.31614/cmes.2019.04376Search in Google Scholar
[37] Mohammadi R . Exponential Bspline solution of convection–diffusion equations. Appl Math. 2013;4:933–44. 10.4236/am.2013.46129Search in Google Scholar
[38] Mohebbi A , Dehghan M . Highorder compact solution of the onedimensional heat and advection–diffusion equations. Appl Math Model. 2010;34:3071–84. 10.1016/j.apm.2010.01.013Search 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 . Spacetime radial basis functions. Comput Math Appl. 2002;43:539–49. 10.1016/S08981221(01)003042Search 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 Bsplines approach. Appl Math Model. 2016;40:4586–611. 10.1016/j.apm.2015.11.041Search in Google Scholar
[42] Parand K , Rad JA . Kansa method for the solution of a parabolic equation with an unknown space wisedependent coefficient subject to an extra measurement. Comput Phys Commun. 2013;184:582–95. 10.1016/j.cpc.2012.10.012Search 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. 10.1007/s4099501703796Search in Google Scholar
[44] Smolianski A , Shipilova O and Haario H . A fast highresolution algorithm for linear convection problems: particle transport method. Int J Numer Methods Eng. 2007;70:655–84. 10.1002/nme.1899Search 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. 10.1016/j.amc.2012.07.027Search in Google Scholar
[47] Zhu XL , Rui HX . Highorder compact difference scheme of 1D nonlinear degenerate convection–reaction–diffusion equation with adaptive algorithm. Numer Heat Tr BFund. 2019;75:43–66.10.1080/10407790.2019.1591858Search 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.