Research Article | Open Access
Numerical Solutions of Two-Way Propagation of Nonlinear Dispersive Waves Using Radial Basis Functions
We obtain the numerical solution of a Boussinesq system for two-way propagation of nonlinear dispersive waves by using the meshless method, based on collocation with radial basis functions. The system of nonlinear partial differential equation is discretized in space by approximating the solution using radial basis functions. The discretization leads to a system of coupled nonlinear ordinary differential equations. The equations are then solved by using the fourth-order Runge-Kutta method. A stability analysis is provided and then the accuracy of method is tested by comparing it with the exact solitary solutions of the Boussinesq system. In addition, the conserved quantities are calculated numerically and compared to an exact solution. The numerical results show excellent agreement with the analytical solution and the calculated conserved quantities.
Consider the initial and boundary value problem where , and are real constants and subscripts and denote space and time derivatives, respectively. In fluid mechanics, the functions and represent flow velocities. Solutions of this type of systems have attracted much research in the past two decades [1–9]. In these studies the most popular system of Boussinesq type is the one proposed by Bona and Chen in , to describe approximately the two-dimensional propagation of surface waves in a uniform horizontal channel of a fixed length filled with an irrotational, incompressible, and inviscid flow. The system is derived formally from Euler’s equations in 2D and using small amplitude and long wave length assumptions. Further, solitary wave solutions of this system have been reported in numerous works; see for instance [1, 2, 6, 10].
This work studies the numerical solution of this system by means of radial basis functions (RBFs). The use of these types of basis functions has become very popular in recent times; see for instance the work of Buhmann , Franke and Schaback , Driscoll and Heryudono , and the references therein. The main inspiration for this work is the very successful application of RBFs to solve the Kawahara equation  and the modified regularized long wave equation (MRLW) .
We will examine the two cases for the Boussinesq type system. The first one we examine is the Bona and Chen system, which is important in the theory of dispersive waves, This corresponds to the case with . The second case we examine is the full system with , , , and : The paper is organized in three sections. In Section 2, we use RBFs to reduce the PDE system to a system of differential equations. In Section 3, we use a linear multistep method to solve these ODEs and then proceed to compute several conserved quantities of the PDE system. We provide comparative tables in order to show the accuracy of the RBFs approach. Finally, in Section 4, we discuss some conclusions.
2. The Method of Lines for Radial Basis Functions
The RBFs approach is one of the most effective meshless methods to solve numerically PDEs. The key feature of the RBFs is that its implementation does not require a mesh at all. The method employs approximants whose values depend only on the distance between some center point and a domain point, in an appropriate norm. Due to the use of the distance functions, we can control the convergence up to an order proportional to the spacing of two points where we evaluate such functions .
The functions and in (1) and (2) are approximated using -terms RBFs: where and are time-dependent functions to be determined and is a radial basis functions to be specified. We adopt the Euclidean -norm in this work to denote distance between the point , referred as center of the RBFs, and , the free spatial variable, at which we can evaluate the approximant. Some examples of the RBFs are [11, 15] where the parameter is proportional to the amplitude of the RBF at , and it is under our control . These RBFs have shown to be successful in solving nonlinear PDEs with mixed partial derivatives [15, 17].
In terms of the RBFs, we can find the derivatives of the approximate solution of terms as follows: as well as for .
We solve the system over the real line and on a finite time . We first truncate the spatial interval to a finite computational domain and specify collocation points . We then employ these equally-spaced collocation points , ; and introduce the matrix notation the vector superscript denotes the transpose. We then obtain a ODE system that can be compactly written in the form where is called a mass matrix, is called the drift matrix, and is in general a nonlinear term dependent on and . The vector is defined by . Explicitly, we have The elements of must be explained. The spatial derivative (15), , with or , is acting element by element on a matrix and does not modify its size. The products and are vectors of size each; however, its product is element by element according to the right-hand side of (12), after the expression is evaluated in a particular collocation point . Therefore, and are vectors of size .
The classical RK4 algorithm for a system of ODEs is given by where
The RK4 scheme does not have stability issues as long as the time step is chosen sufficiently small. The selection of the time step is done by the following .
Rule of Thumbs. The method of lines is stable if the eigenvalues of the linearized spatial discretization operator, scaled by , lie in the stability region of the time-discretization operator.
For the stability analysis nonlinearity in (22), is set to zero and we compute the eigenvalues of . So the method is stable as long as the eigenvalues are in the stability region (). In order to have a stable scheme, we adjust the parameter so that the matrices remain in the region of stability. This was carried out in the numerical experiments.
3. Numerical Approximations of Boussinesq System
In this section, we use the proposed algorithm to calculate the numerical solution of the Boussinesq type system and its conserved quantities. These quantities were performed by means of the trapezoidal quadrature. The accuracy of the method is tested by computing the and (error norms) defined as where and stand for numerical evaluation of the exact solution and numerical approximation of the solution, respectively. The test cases in this section are solved over the domain in the time period . The domain and time period are discretized with values and , respectively. The initial conditions are taken from the exact solution at the value .
The integrals involving the conserved quantities were calculated by using the Trapezoidal rule.
3.1. Numerical Benchmark
The first test problem was studied extensively by Chen in . This case is important in the theory dispersive waves. A solitary wave solutions is known and is given by Chen in : This case has four conserved quantities derived in : In Tables 1 and 2 we have the error norms at time for and , respectively. We also measure the amplitude for and the peak-to-peak (P-P) amplitude for . Table 3 shows the analytical and numerical value of the conserved quantities. The tables show excellent agreement with the exact solution. We can see that using multiquadratic and Gaussian quadratic RBFs is more accurate in approximating the solution which is in agreement with the work of Dereli  for MRLW equation. In addition to these tables, we see in Figure 1 how the approximate solution is “lying” closely to the analytical one. Figure 2 shows that as we increase the number of collocation points the error reduces.
(a) Evolution of
(b) Evolution of
In this example we test our algorithm with a solution against the “full system” equations (7). A solitary wave solution to the “full system” is presented in . The solution is as follows: The boundary conditions are the same as the BBM and the conserved quantities are Tables 4, 5, 6, and 7 show the and error for various times of and along with the conserved quantities of the equations. Once again the numerical solution and the analytical solution are in good agreement.
In this last example, we present the interaction between two solitons. The initial