Analysis of New RBF-FD Weights, Calculated Based on Inverse Quadratic Functions

Rahimi, Asghar; Shivanian, C.A.Elyas; Abbasbandy, Saeid

doi:https://doi.org/10.1155/2022/3718132

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Research Article | Open Access

Volume 2022 | Article ID 3718132 | https://doi.org/10.1155/2022/3718132

Analysis of New RBF-FD Weights, Calculated Based on Inverse Quadratic Functions

Asghar Rahimi,¹C.A.Elyas Shivanian,¹and Saeid Abbasbandy¹

Academic Editor: Mubashir Qayyum

Received18 Feb 2022

Accepted09 Apr 2022

Published28 Apr 2022

Abstract

Local radial basis functions (RBFs) have many advantages for solution of differential equations. In some of these radial functions, there is a parameter that has a special effect on the accuracy of the answer and is known as the shape parameter. In this article, first of all, we derive inverse quadratic (IQ)-based RBF-generated finite difference coefficients for some derivatives in one dimension (1D). Then, to evaluate the efficiency of these new weights and also the effect of the shape parameter on the accuracy of the resulting approximations, we will test them with a suitable function. After that, we focus on solving some boundary value problems (BVPs), using IQ-based RBF-FD method. There is a range for the shape parameter in which the approximation error is less than other areas. We use an efficient algorithm to find the best value of the RBF parameter for the problem domain. Our studies show that IQ-based RBF-FD weights could be derived analytically easier than multiquadrics (MQs) which were previously presented in the literature. Besides, the results of numerical examples confirm the high accuracy of these new formulas. For better comparison, we revisit some previously studied illustrative examples.

1. Introduction

Finite difference has been one of the main methods for solving partial differential equations (PDEs), maybe from 1911 [25] up to now. Easy implementation is an attractive feature of these methods, but less geometrical flexibility is a serious drawback. Although Rolland Hardy, in 1971 [18], recommended the radial basis function approach for interpolating scattered 2D data, in 1990 [20, 21], Kansa realized that the RBFs could be used as a tool to approach the function derivatives, and therefore he opened a new window for solving partial differential equations, numerically.

In the past years, global RBFs have been considered as a method for numerically solving PDEs. One of the most important features of these methods is spectral accuracy, but they generally lead to a large, ill-conditioned, dense linear system which should be solved. These computational costs have been an important drawback with global RBFs and a strong motivation to construct a new meshfree method. This young method which is called RBF-FD is based on the famous finite difference scheme and could be implemented on unstructured node distributions and irregular domains, through using RBFs [10, 15, 16, 26, 33].

RBF-FD methods came into view through a conference presentation by Tolstykh in 2000 [30]. In 2003, two important works proposed by Shu et al. [28] and again by Tolstykh et al. [31] attracted more attention on the RBF-FD methods. These local methods relinquish spectral accuracy but achieve a sparse and better conditioned linear system [32, 33].

One of the common ways to calculate differencing coefficients in the classical FD method is the use of polynomials. Tolstykh in [31] says that the key concept in the RBF-FD method is to choose RBF interpolants to approximate derivatives in local domains using some neighbor nodes. Therefore, in the RBF-FD method, like the FD, some molecular shapes or clouds would be considered. Here, we use the word for such clouds.

Generally, RBF-FD methods contain the following steps:(i)Node generation.(ii)Defining for every node in the problem domain, a stencil with a number of neighboring nodes.(iii)Constructing an approximation for each differential operator, using a linear combination of the values of the unknown function at the nodes scattered in the stencil.(iv)Computing or differencing coefficients for each stencil.(v)Substituting the earned approximations in step (iv) for derivatives, at each node in the PDE, to earn the corresponding final system.(vi)Solving the global system.

Different node generations, different RBFs, different stencils, and also different ways to find neighbor nodes in the stencil could affect the efficiency of the method. There has always been the important question of which radial function in any situation should be used to get the best result. If in the method we use the RBFs which contain a shape parameter, its value should be definite before computing weights. The exactness of the approximation is very dependent on choosing suitable value for the shape parameter.

Several works have been conducted related to the shape parameter like Fassauer and Zhang [11] and Fornberg et al. [14]. Bayona et al. in [1] at first derived MQ-based RBF-FD formulas and then using these formulas in [2, 3] recommended two different algorithms for choosing the shape parameter. In [2], they introduced an algorithm which gives a global shape parameter. It means for all of the problem domain, just a single value for the shape parameter is used. In another work, they proposed a way for finding optimal shape parameter for each node, i e., a local or node dependent algorithm was used there. There are some other works relating to the RBF-FD methods using MQs, Gaussians (GA), and inverse multiquadrics (IMQs) [4, 13, 29], but we could not find any work relating to the local inverse quadratic-based RBF-FD method.

So, in this paper, in the next section, we briefly introduce our method. Then in Section 3, we derive IQ-based RBF-FD formulas, and after that, using these formulas, in Section 4, we construct our RBF-FD method and solve some BVPs. In this step, we use the algorithm proposed in [2] for finding optimal constant shape parameter. At the end, main conclusions and future work suggestions are included.

The RBF-FD method is still developing, and several researchers are working on different aspects of the method. For more studies about the RBF-FD method, the interested readers are referred to [5–9, 12, 15, 17, 19, 22–24, 27] and references therein.

2. A Review of RBF-FD Method and Some Background Information about Approximation Error

Consider the Dirichlet BVP:in which the differential operator would be approximated at the node , the computational domain is considered bounded, and the real functions and are defined in and on , respectively. Consider a stencil formed by nodes . These stencils could be structured or unstructured. They even could be in different shapes for each node. In this paper, our analysis will focus on structured stencils. For approximating at the node , we will use a linear combination of the values of at these nodes, i.e., and nodes surrounding it. In other words, we can write the formulawhere are the unknown weights, which should be computed. In traditional FD formulas, one can compute the unknown weights by polynomial interpolation. As mentioned before, we will use RBFs instead of polynomials here. Then, we havewhere is the standard Euclidean distance between the point of interest and a node at and is a RBF containing the free shape parameter .

Substituting (3) into (2) and doing some algebra yields a linear system as follows. By solving this system, we will obtain the unknown weights .

Resulting coefficients rely on just which is the spacing factor and . In the consequent relations, we suppose is known. Hence, the weights will be dependent only on . For discretization of the domain of problem, we use generally nodes, in which nodes of them are located on the boundary of the and nodes are in . Approximation of the using the relation (2) yields a local RBF-FD error, and we use the notation for it. Therefore, if we replace (2) into (1), for a node in , we can write

The equations in (5) could be rewritten as follows:

In the foregoing equation, is a square matrix with the dimension of . This matrix is sparce and its entries are the weights . The vector collects the exact solutions at the interior nodes. The values of in (1), at each interior node, are gathered in the vector , and finally, the notation is assigned for the vector of local approximation errors for the all interior nodes.

Solving the linear system from (6), we can gain our approximation of which is shown by as follows:and hence the corresponding RBF-FD error would be of the form

Now, we should search for a special for which error (8) should be minimum with respect to a norm. is nominated here as optimal constant shape parameter and assigned with . Consequently, using infinite norm, we have

Because the exact solution is unknown in practice, (9), could not be used directly for finding . So, we use (6) and (7), and thus

The local errors , which will be approximated analytically in Section 3, are as series expansions in powers of . These formulas are functions of and and also contain and its derivatives. Then, to estimate and its derivatives, we will use the finite difference approximation , and hence we are able to find the value of , which stands for an estimation of optimal shape parameter . Therefore,in which is an estimation of .

3. Deriving New IQ-RBF-FD Coefficients and Local Error Estimation

In this paper, we use inverse quadratics (IQs) as radial basis functions, and thus

For constructing IQ-based RBF-FD formulas to approximate the differential operator , we first use (4) to compute the unknown weights. As mentioned before, these weights are functions of and . They are obtained analytically using Mathematica. Then, we use them to derive analytical expressions for local truncation error. In a stencil consisting of nodes, the local truncation error for the node is defined as

It is notable that we use the weights in the form of Taylor series expansions in powers of . For confirming the relations obtained in Sections 3.1 and 3.2, we employ which is used in [1] as test function for MQ-based RBF-FD formulas.

3.1. IQ-RBF-FD Weights and Error Analysis for

The IQ-RBF-FD formulas to approximate in 1D are sorted in Table 1. Series expansions are given for three, five, and seven equispaced nodes. Using the weights of three-node stencil in (13) and expanding and , we find the corresponding local truncation error

Similarly, for five and seven nodes, we have

In Figure 1, corresponding errors for three, five, and seven nodes (i.e., equations (14)–(16)) are compared. The figure shows absolute value of the errors as a function of for at node . It is clear from the figure that error decreases when the number of nodes in the stencil increases. Notice also that there is a range of values of , for which the IQ-RBF-FD formulas are more accurate. So, these error formulas could be used to find a specific , which leads to the minimum error.

3.2. IQ-RBF-FD Weights and Error Analysis for

Table 2 shows the weights for IQ-RBF-FD formulas to approximate in 1D. As in the first derivative, series expansions are given for three, five, and seven equispaced nodes. The corresponding errors are

Figure 2 is the same as Figure 1, but for the errors of approximating . Again, we see that the error decreases for increasing , and also there is a range of for which the IQ-RBF-FD formulas are more accurate.

We mentioned before that to obtain closed-form solutions for the coefficients and for the error, we use Mathematica. It is worth to say that including extra nodes into stencil makes the computational requirements to be high. However, our studies show that using IQs leads to less computational costs than MQs.

4. The Method Implementation on Some Boundary Value Problems

At first, using structured nodes, we perform our IQ-RBF-FD method on two BVPs in 1D. Because the algorithm here is the same as in [2], we employ the same examples for better comparison. Our investigations indicate that applying the optimal shape parameter which is estimated based on IQ RBFs leads to better solutions than MQs. We will then compare the effect of IQ RBFs on a two-dimensional example in both uniform and scattered distributions.

4.1. First Example

Consider the following Dirichlet BVP:where the exact solution is and is computed accordingly. For discretization of the domain in (19), we use structured nodes. To approximate the second derivative, let us use a three-node approximation. So, the local error would be (17). As mentioned before, because in real situations we do not know the exact solution, using a classical FD approximation, we estimate (17). Therefore, in (17), we use a second-order central difference approximation of , which is shown with the notation . Besides, and could be computed from the right hand side of (19), accordingly. So, we have

Using (20) helps us to estimate the local approximation error at each interior node, and then the vectorcould be obtained.

Besides, using (4), we compute weighting coefficients which are entries of the sparse matrix . Now, applying (11), we can find . For this goal, i.e., estimating a specific which minimizes , we use Matlab command . At the end, we are able to achieve the optimal IQ-RBF-FD approximation of as follows:

In our numerical studies, to show the abilities of the new IQ-RBF-FD method, the infinite norm error alongside the following root mean square (RMS) error is used:where and are achieved by exact and approximate solutions on points and is the number of interior nodal points.

In Table 3, first of all, we have reported the infinite norm of the classical FD solution error. After that, and its related IQ-RBF-FD errors could be seen. The results compared to the classical FD are very accurate. Besides, our IQ-based method leads to better errors than the MQ-based method in [2].

4.2. Second Example

Consider the following steady convection-diffusion problem:where the exact solution is . Like the first problem, we use three-node approximation for the convection-diffusion differential operator. Consequently,

In the foregoing formula, same as the first example, we use a second-order central difference scheme to achieve as an approximate solution for exact . Then, using , we estimate through the second-order central difference scheme. For higher derivatives, we use the following relation:in which is the right hand side of (24).

Doing the same steps mentioned in the first example, we obtain the information as shown in Table 4. Similar to problem (19), the results of IQ-RBF-FD are very exact and better than the errors reported in [2] for MQ-RBF-FD.

4.3. Testing the Method on a 2D Example

In the present paper, our goal is not to study two-dimensional problems, but to see the applicability of the method for problems with higher dimensions and different node distributions, so we test the use of IQ functions in both uniform and scattered node distribution modes on a 2D boundary value problem. Consider a 2D Poisson problem as follows:in which is the Laplace linear differential operator and and are computed using the known exact solution:and the problem domain is

The results of employing the method on problem (27) for both structured and unstructured node distributions are reported in Table 5. It is remarkable that for the unstructured node generation, Halton points are used. In the table, N represents the number of total nodes and c represents the shape parameter of the IQ functions. The stencils used in the structured and unstructured modes contained 5 and 11 nodes, respectively. It could be seen that with the increase of the number of total nodes, the accuracy of the answers improves.

5. Conclusion

In this paper, we use inverse quadratic (IQ) radial basis function for deriving new RBF-FD formulas. We compute IQ-RBF-FD weighting coefficients analytically, easier than MQ-RBF-FD weights in [1] for higher nodes. Then, using these weighting coefficients, we approximate first and second-order one-dimensional derivatives. To show the efficiency of these new formulas, we test them on a test function and achieve very good results. After that, based on IQ-RBF-FD formulas, we propose our method and implement it on some boundary value problems. IQ radial basis functions contain a free shape parameter which is very important for accuracy of the approximation. The main challenge of the current work is to find an optimal shape parameter, for which the approximation error should be minimum. This is done based on an efficient algorithm proposed in [2]. We have obtained very accurate results here and also better errors than the MQ-RBF-FD method proposed in [2]. It seems to be a good idea to derive IQ-based RBF-FD formulas for higher dimensions and different node distributions and implement them to other problems in future works.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

V. Bayona, M. Moscoso, M. Carretero, and M. Kindelan, “RBF-FD formulas and convergence properties,” Journal of Computational Physics, vol. 229, no. 22, pp. 8281–8295, 2010.
View at: Publisher Site | Google Scholar
V. Bayona, M. Moscoso, and M. Kindelan, “Optimal constant shape parameter for multiquadric based RBF-FD method,” Journal of Computational Physics, vol. 230, no. 19, pp. 7384–7399, 2011.
View at: Publisher Site | Google Scholar
V. Bayona, M. Moscoso, and M. Kindelan, “Optimal variable shape parameter for multiquadric based RBF-FD method,” Journal of Computational Physics, vol. 231, no. 6, pp. 2466–2481, 2012.
View at: Publisher Site | Google Scholar
V. Bayona, M. Moscoso, and M. Kindelan, “Gaussian RBF-FD weights and its corresponding local truncation errors,” Engineering Analysis with Boundary Elements, vol. 36, no. 9, pp. 1361–1369, 2012.
View at: Publisher Site | Google Scholar
V. Bayona, N. Flyer, B. Fornberg, and G. A. Barnett, “On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs,” Journal of Computational Physics, vol. 332, pp. 257–273, 2017.
View at: Publisher Site | Google Scholar
V. Bayona, “An insight into RBF-FD approximations augmented with polynomials,” Computers & Mathematics with Applications, vol. 77, no. 9, pp. 2337–2353, 2019.
View at: Publisher Site | Google Scholar
V. Bayona, N. Flyer, and B. Fornberg, “On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries,” Journal of Computational Physics, vol. 380, pp. 378–399, 2019.
View at: Publisher Site | Google Scholar
M. Dehghan and V. Mohammadi, “A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge–Kutta method,” Computer Physics Communications, vol. 217, pp. 23–34, 2017.
View at: Publisher Site | Google Scholar
H. Ding, C. Shu, and D. B. Tang, “Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments,” International Journal for Numerical Methods in Engineering, vol. 63, no. 11, pp. 1513–1529, 2005.
View at: Publisher Site | Google Scholar
G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishers, Singapore, vol. 6, 2007.
G. E. Fasshauer and J. G. Zhang, “On choosing “optimal” shape parameters for RBF approximation,” Numerical Algorithms, vol. 45, no. 1-4, pp. 345–368, 2007.
View at: Publisher Site | Google Scholar
N. Flyer, B. Fornberg, V. Bayona, and G. A. Barnett, “On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy,” Journal of Computational Physics, vol. 321, pp. 21–38, 2016.
View at: Publisher Site | Google Scholar
B. Fornberg and G. Wright, “Stable computation of multiquadric interpolants for all values of the shape parameter,” Computers & Mathematics with Applications, vol. 48, no. 5-6, pp. 853–867, 2004.
View at: Publisher Site | Google Scholar
B. Fornberg and C. Piret, “On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere,” Journal of Computational Physics, vol. 227, no. 5, pp. 2758–2780, 2008.
View at: Publisher Site | Google Scholar
B. Fornberg and N. Flyer, A Primer on Radial Basis Functions with Applications to the Geosciences, SIAM, Philadelphia, USA, 2015.
B. Fornberg and N. Flyer, “Solving PDEs with radial basis functions,” Acta Numerica, vol. 24, pp. 215–258, 2015.
View at: Publisher Site | Google Scholar
D. Gunderman, N. Flyer, and B. Fornberg, “Transport schemes in spherical geometries using spline-based RBF-FD with polynomials,” Journal of Computational Physics, vol. 408, Article ID 109256, 2020.
View at: Publisher Site | Google Scholar
R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces,” Journal of Geophysical Research, vol. 76, no. 8, pp. 1905–1915, 1971.
View at: Publisher Site | Google Scholar
H. Jafari and F. Fakhr Kazemi, “Stable RBF-RA method for solving fuzzy fractional kinetic equation,” International Journal of Dynamical Systems and Differential Equations, vol. 12, no. 2, p. 1, 2022.
View at: Publisher Site | Google Scholar
E. J. Kansa, “Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates,” Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 127–145, 1990.
View at: Publisher Site | Google Scholar
E. J. Kansa, “Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics II: solutions to parabolic, hyperbolic and elliptic partial differential equations,” Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 147–161, 1990.
View at: Publisher Site | Google Scholar
O. Nikan, H. Jafari, and A. Golbabai, “Numerical analysis of the fractional evolution model for heat flow in materials with memory,” Alexandria Engineering Journal, vol. 59, no. 4, pp. 2627–2637, 2020.
View at: Publisher Site | Google Scholar
D. T. Oanh, O. Davydov, and H. X. Phu, “Adaptive RBF-FD method for elliptic problems with point singularities in 2D,” Applied Mathematics and Computation, vol. 313, pp. 474–497, 2017.
View at: Publisher Site | Google Scholar
J. Rashidinia and M. N. Rasoulizadeh, “Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation,” Wave Motion, vol. 90, pp. 152–167, 2019.
View at: Publisher Site | Google Scholar
L. F. Richardson, “The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam,” Philosophical Transactions of the Royal Society of London, vol. 210, pp. 307–357, 1911.
View at: Google Scholar
R. Schaback, “Error estimates and condition numbers for radial basis function interpolation,” Advances in Computational Mathematics, vol. 3, pp. 251–264, 1995.
View at: Publisher Site | Google Scholar
V. Shankar, G. B. Wright, R. M. Kirby, and A. L. Fogelson, “A radial basis function (RBF)- finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces,” Journal of Scientific Computing, vol. 63, no. 3, pp. 745–768, 2015.
View at: Publisher Site | Google Scholar
C. Shu, H. Ding, and K. S. Yeo, “Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 7-8, pp. 941–954, 2003.
View at: Publisher Site | Google Scholar
F. Soleymani, M. Barfeie, and F. K. Haghani, “Inverse multi-quadric RBF for computing the weights of FD method: application to American options,” CommuNications in Nonlinear Science and Numerical Simulation, vol. 64, pp. 74–88, 2018.
View at: Publisher Site | Google Scholar
A. I. Tolstykh, “On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations,” Proceedings of the 16th IMACSWorld Congress, Lausanne, vol. 228, pp. 4606–4624, 2000.
View at: Google Scholar
A. I. Tolstykh and D. A. Shirobokov, “On using radial basis functions in a “finite difference mode” with applications to elasticity problems,” Computational Mechanics, vol. 33, no. 1, pp. 68–79, 2003.
View at: Publisher Site | Google Scholar
G. B. Wright, Radial Basis Function Interpolation: Numerical and Analytical Developments, University of Colorado, Boulder, CO, 2003, Ph.D. thesis.
G. B. Wright and B. Fornberg, “Scattered node compact finite difference-type formulas generated from radial basis functions,” Journal of Computational Physics, vol. 212, no. 1, pp. 99–123, 2006.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Asghar Rahimi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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