# Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

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## Abstract

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.

## Keywords

Collocation method Radial basis function Fictitious point Pseudospectral Resampling Rosenau equation Multiple boundary conditions## Mathematics Subject Classification

65M70 35G31## 1 Introduction

*solitons*are colliding with one another, is indispensable in digital transmission through optical fibers. As data carriers, we need solitons that interact “cleanly” in the sense that none of the solitons loose any information, shape, or other conserved quantities, when they pass through each other. One may consult [7] for a fascinating history behind this subject. The Rosenau equation in its general form is given by

*g*(

*u*) is polynomial of degree \(q+1\), \(q\ge 1\). Multiple boundary conditions are required at the boundary \(\partial \varOmega \), such as

*n*is the outward normal direction from the boundary, and we need an initial condition

The objective of this paper is to derive numerical methods based on radial basis function (RBF) collocation methods [14, 22] for the Rosenau equation, that can be applied to problems in one, two, and three space dimensions, for non-trivial geometries. These methods will also be applicable to other higher order partial differential equations. We derive and implement a fictitious point RBF (FP–RBF) collocation method and a resampling RBF (RS–RBF) collocation method, and perform experiments in one and two space dimensions. We investigate the accuracy and behavior of the derived methods theoretically and numerically. We also compare the RBF methods with pseudospectral (PS) methods [12, 35] with respect to accuracy in one space dimension.

In this paper we are using global RBF approximation as a test case for implementation of multiple boundary conditions in general geometries. The current direction in the research on RBF approximation methods for PDEs is towards the use of localized RBF approximation methods. The main categories are stencil-based methods (RBF-FD) [2, 11] and partition of unity methods (RBF–PUM) [23, 32]. The (FP–RBF) technique should carry over in both cases, with minor differences in the implementation, whereas the (RS–RBF) method should be applicable to RBF–PUM, but not as easily to RBF-FD.

The outline of the paper is as follows: In Sect. 2, a basic RBF collocation scheme is introduced. Section 3 describes different approaches to handle the multiple boundary conditions. Then in Sect. 4 the theoretical approximation properties of the RBF method for the one-dimensional Rosenau problem are discussed, while the details of the analysis are given in Appendix A. The implementation aspects are discussed in Sect. 5, followed by numerical results in Sect. 6. Finally, Sect. 7 contains conclusions and discussion.

## 2 The Basic RBF Collocation Scheme

*A*is non-singular. This holds for commonly used RBFs such as Gaussians, inverse multiquadrics and multiquadrics [25, 33] for distinct node points \(x_j\). We can furthermore, use (2.3) to identify cardinal basis functions such that we can write the approximation on the FEM like form

*u*(

*x*,

*t*) is approximated by

## 3 Dealing with Multiple Boundary Conditions

### 3.1 Transforming to Lower Order System

*u*at the boundaries become Dirichlet conditions for

*w*. However, the system to solve becomes twice as large, as we need a total of 2

*N*degrees of freedom for

*u*and

*w*. Due to this reason, and especially for global RBFs where differentiation matrices are dense, we are not pursuing this method. However, for localized RBF methods, where differentiation matrices are sparse, this method may still be worth trying.

### 3.2 Fictitious Point Method

Fictitious or ghost point methods have been commonly used as a way to enforce multiple boundary conditions in finite difference methods. The implementation for global collocation methods such as pseudospectral methods is due to Fornberg [13].

Let \(-L=x_2<x_3<\cdots < x_{N-1}=L\) be distinct node points. The Dirichlet conditions (1.2) can be imposed by fixing the values for \(u(x_2)\) and \(u(x_{N-1})\), but for the Neumann conditions (1.3) we use the fictitious point approach proposed by Fornberg [13], and introduce two additional points at some arbitrary locations denoted by \(x_1\) and \(x_{N}\).

*s*(

*x*,

*t*) as in (2.7), extended to include the fictitious points, for the spatial approximation of the solution

*u*(

*x*,

*t*),

*F*(

*x*,

*t*) can be directly identified from (3.6). In this simple two point boundary case, we can actually derive the explicit form of the modified basis for illustration. This yields

The mass matrix *Q*(*t*) is in general invertible but non-singularity cannot be guaranteed. Kansa [20] argued that if the centers of the RBFs are distinct and the PDE problem is well-posed, matrices discretizing spatial operators are generally found to be non-singular. Hon and Schaback [18] showed that occurrences of singular matrices are very rare, but do exist. When \(\alpha (x,t)\) is constant, the mass matrix is constant over time. Then we can LU-factorize the matrix once and use this factorization throughout the time stepping algorithm.

### 3.3 Resampling Method

*u*(

*x*,

*t*) be approximated in Lagrange form by

*N*unknown functions, and we have used four equations for the boundary conditions. This means that we need \(N-4\) additional equations. The idea in the resampling method is to collocate the PDE at \(N-4\) auxiliary interior points, instead of collocating at the node points. We define the auxiliary points \(\tilde{x}_i\), \(i=1,\ldots ,N-4\) and collocate the PDE to get the equations

The system of equations (3.19) can be solved using a differential algebraic solver. See DASPK [3, 29]. An example of how this can be implemented in MATLAB is given in Sect. 5.

### 3.4 Generalization to More Space Dimensions

The main differences when moving to more than one space dimensions is that we have a boundary curve or a boundary surface that is discretized in the same way as the interior of the domain instead of just two boundary points. The formulation of the two methods is in all essential parts the same, and the formulations (3.13) and (3.19) are valid in the same form, but when we before had two boundary points and two fictitious points, we instead have \(N_b\) boundary points and \(N_b\) fictitious points. Similarly, for the resampling method, we have \(2N_b\) boundary conditions, and therefore we collocate the PDE at \(N-2N_b\) auxiliary points. Experiments for problems in two space dimensions are presented in Sect. 6.

## 4 Error Estimates

*r*(

*t*) represents a polynomial growth factor in time. The function \(C_3(t)\) becomes large due to the matrix norms. When estimated separately like this, the product \(\Vert Q^{-1}\Vert \Vert \tilde{\varPsi }_x\Vert \) becomes large, growing as \(\mathscr {O}(h^{-1})\) (cf. Sect. A.5). The way we have derived the estimates does not easily allow for an estimate that takes the norm of the product \(\Vert Q^{-1}\tilde{\varPsi }_x\Vert \) instead. However, this is in principle the way the matrices appear in the critical terms, and if the product norm is investigated numerically, it turns out to be small.

Because of this overestimate of the time growth, the error estimates are not quantitatively useful, but they provide a qualitative insight into how the different errors interact, and illustrate the difficulties of bounding the non-linear terms in an effective way.

## 5 MATLAB Implementation

In this section, sample MATLAB implementations of the fictitious point and resampling RBF methods for the one-dimensional Rosenau equation (3.1)–(3.3) are presented and discussed. We use an example with a known solution. For \(g(u)=10u^3-12u^5-\frac{3}{2}u\) and \(\alpha (x,t)=0.5\) it holds that \(u(x,t)= sech (x-t)\) is a solution [28]. For both methods, equally spaced nodes are used, and the spatial domain is \([-L,L]\).

### 5.1 Implementation of the Fictitious Point Method

**ode15s**. The two functions below constitute a complete MATLAB implementation of the problem.

It can be noted that when we use the modified basis functions, we need to provide the time derivatives of the boundary conditions as well as the boundary conditions themselves. This is not needed with the alternative formulation (3.13), but instead the system is stated in DAE form.

### 5.2 Implementation of the Resampling RBF Method

**ode15s**, previously employed for the fictitious point method. One may also use the syntactically similar open source software OCTAVE and there use

**dasspk**as DAE solver. The following two functions show the MATLAB implementation of the resampling RBF method.

## 6 Numerical Results

In this section, we perform numerical experiments to investigate the accuracy and convergence of the proposed schemes. Both one-dimensional and two-dimensional test cases are considered. In all tests, the inverse multiquadric RBF is used. The shape parameter values are chosen using a parametric relation such that they fall within the region where the method is stable, while making the solution as accurate as possible. For the one-dimensional test case, we compare the results with those of a pseudo-spectral resampling (RS–PS) and a pseudo-spectral fictitious point (FP–PS) method. We have not included the code here, but it can be downloaded from the authors’ web pages.

### 6.1 The One-Dimensional Case

We consider the same test problem as in the sample implementations in Sect. 5, with \(g(u)=10u^3-12u^5-\frac{3}{2}u\) and known exact solution \(u(x,t)=\text {sech}(x-t)\). Equispaced node distributions over the interval \([-L,\,L]\) are used for the RBF methods. The total number of points *N* includes the fictitious points in the FP–RBF case. The initial and boundary conditions are taken from the exact solution.

*t*together with the numerical solution using the FP–RBF method in Fig. 2. The solution is a pulse that travels to the right with time.

*L*are plotted in Fig. 4. As shown in the figure, the errors of both the RBF based methods and the pseudo-spectral methods are similar in magnitude. For the shorter interval, the RS–PS method has smaller errors near the boundaries, which is consistent with the clustering of the Chebyshev nodes. However, for the larger interval, where the solution is small at the boundary, this effect is not visible.

*N*for the two RBF methods compared with the PS methods. For all four methods, the highest attainable accuracy is almost the same. When \(\varepsilon ^{}h\) is constant, as in this experiment, we would expect the error to reach a saturation level, but accuracy is also limited by conditioning, and this may be the effect that we see here. In both cases, the FP–RBF method reaches the highest accuracy faster than the RS–RBF method. The PS methods performs best for the short interval, and performs worse than the RBF methods for the longer interval. One explanation for this can be that the node density for the Chebyshev nodes compared with the uniform nodes is lower in the interesting region (middle of the domain) in this case.

### 6.2 Two-Dimensional Square Domain

*h*. In Fig. 9, to make a fair comparison, we plot the error as a function of \(\sqrt{N}\propto 1/h\). For this range of

*N*-values, the conditioning is low enough to not influence the error, and exponential convergence can be observed for both RBF methods. We see that the estimated slopes in the right subfigure are precisely double those in the left subfigures. If we take into account that

*h*is also twice as large for the case \(L=2\), we can conclude that the rate of convergence in terms of 1 /

*h*is the same in both cases.

### 6.3 Two-Dimensional Irregular Domain with Smooth Boundary

*N*is the total number of points, where the number of fictitious points outside the domain is \(N_b\), i.e., \(N=N_d+2N_b\), and the number of auxiliary points inside the domain in the resampling method is \(N_d-2N_b\).

Overall, the error trends are similar to those for the square domain, showing that the RBF methods provide a well functioning generalization of both the fictitious point method and the resampling method to general domains.

## 7 Conclusion

The Rosenau equation, which is used as an application throughout this paper, is an example of a non-linear PDE with multiple boundary conditions as well as mixed space-time derivatives. Multiple boundary conditions provides an extra challenge when solving PDE problems. The standard form of a typical collocation method assumes that one condition is imposed at each node point/grid point. Hence, the additional conditions at the boundary nodes lead to a mismatch between the number of conditions and the number of unknowns.

Two approaches to manage multiple boundary conditions that have been introduced for spectral methods are fictitious point methods and resampling methods. In this paper we have shown how to implement these approaches in the context of RBF collocation methods. From numerical experiments for a one-dimensional test problem, we could see that the behavior of the method with respect to accuracy in space and time is very similar to that of the corresponding pseudo-spectral method.

For two-dimensional problems, already in a regular geometry such as the square, the application of spectral methods becomes more complicated. Approximations are typically based on tensor product grids, but if we use the one-dimensional extension techniques for each grid line, again the number of extra conditions and extra points do not naturally match. The problem can for example be resolved by choosing one of the directions for the corner points, but then the approximations in the other direction needs to be of lower order.

We show that with the two RBF methods, due to the freedom of node placement, we can distribute the fictitious points or the resampled nodes uniformly and symmetrically with respect to the domain. Furthermore, we show that the concept can be transferred also to irregularly shaped domains.

We have also analyzed the theoretical properties of the fictitious point RBF approximation for the one-dimensional Rosenau equation. We could show that the spectral convergence of the spatial approximation carries over to the PDE solution, while the growth of the error in time in our estimate strongly depends on the bounds on the non-linear term.

To conclude, both the implemented approaches are promising for problems with multiple boundary conditions, especially for geometries where spectral methods cannot easily be applied. Global RBF approximations as the ones used here are competitive for problems in one or two space dimensions, but the computational cost can become prohibitive for higher-dimensional problems due to the need to solve dense linear systems. Therefore, an interesting future direction is to see how resampling and fictitious point methods can be combined with localized (stencil or partition based) RBF methods.

## Supplementary material

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