Magnetohydrodynamic simulations using radial basis functions

Abstract

To overcome the computational mesh quality difficulties, mesh-free methods have been developed. One of the most popular mesh-free kernel approximation techniques is radial basis functions (RBFs). Initially, RBFs were developed for multivariate data and function interpolation. It is well-known that a good interpolation scheme also has great potential for solving partial differential equations. In the present study, the RBFs are used to interpolate stream-function and temperature in a two-dimensional thermal buoyancy flow acted upon by an externally applied steady magnetic field. Use of mesh-free methods promises to significantly reduce the computing time, especially for the complex classes of problems such as magnetohydrodynamics.

Introduction

Radial basis functions are essential ingredients of the techniques generally known as “meshless methods”. In one way or another all meshless techniques require some sort of radial function to measure the influence of a given location on another part of the domain. The use of radial basis functions (RBF) followed by collocation, a technique first proposed by Kansa [1], after the work of Hardy [2] on multivariate approximation, is now becoming an established approach and various applications to problems of structures and fluids have been made in recent years. See, for example, Leitão [3], [4].

Kansa’s method (or asymmetric collocation) starts by building an approximation to the field of interest (normally displacement components) from the superposition of radial basis functions (globally or compactly supported) conveniently placed at points in the domain (and, or, at the boundary).

The unknowns (which are the coefficients of each RBF) are obtained from the (approximate) enforcement of the boundary conditions as well as the governing equations by means of collocation. Usually, this approximation only considers regular radial basis functions, such as the globally supported multiquadrics or the compactly supported Wendland [5] functions.

Radial basis functions (RBFs) may be classified into two main groups:

• the globally supported ones namely the multiquadric (MQ, $\sqrt{(x-x_j{)}^2+c_j^2}\text{,}$ where c_j is a shape parameter), the inverse multiquadric, thin plate splines, Gaussians, etc;

• the compactly supported ones such as the Wendland [5] family (for example, $(1-r{)}_{+}^n+p(r)$ where p(r) is a polynomial and $(1-r{)}_{+}^n$ is 0 for r greater than the support).

In a very brief manner, interpolation with RBFs may take the form:

$$s(x_i)=f(x_i)=\sum _{j=1}^N{\alpha }_j\varphi (|x_i-x_j|)+\sum _{k=1}^{\hat{N}}{\beta }_k{p}_k(x_i)$$

where f(x_i) is known for a series of points x_i and p_k(x_i) is one of the $\hat{N}$ terms of a given basis of polynomials [6]. This approximation is solved for the α_j unknowns from the system of N linear equations, subject to the conditions (for the sake of uniqueness)

$$\sum _{j=1}^N{\alpha }_j{p}_k(x_j)=0\text{.}$$

By using the same reasoning, it is possible to extend the interpolation concept to that of finding the approximate solution of partial differential equations. This is made by applying the corresponding differential operators to the RBFs and then to use collocation at an appropriate set of boundary and domain points. In short, the non-symmetrical collocation is the application of the domain and boundary differential operators LI and LB, respectively, to a set of N–M domain collocation points and M boundary collocation points. From this, a system of linear equations of the following type may be obtained:

$$\begin{array}{cc}\hfill & {\mathit{LIu}}_h(x_i)=\sum _{j=1}^N{\alpha }_j\mathit{LI}\varphi (|x_i-{\epsilon }_j|)+\sum _{k=1}^{\hat{N}}{\beta }_k{\mathit{LIp}}_k(x_i)\hfill \\ \hfill & {\mathit{LBu}}_h(x_i)=\sum _{j=1}^N{\alpha }_j\mathit{LB}\varphi (|x_i-{\epsilon }_j|)+\sum _{k=1}^{\hat{N}}{\beta }_k{\mathit{LBp}}_k(x_i)\hfill \end{array}$$

subject to the conditions ${\sum }_{j=1}^N{\alpha }_j{p}_k(x_j)=0$ where the α_j and β_k unknowns are determined from the satisfaction of the domain and boundary constraints at the collocation points.