Finite Volume Method Based On Radial Basis Functions

In constructing of finite volume scheme, it is a key step to construct the numerical flux terms, which will directly affects the accuracy and resolution of the schemes, especially the adaptive ability of the method. Ordinarily the flux terms were approximated by high order interpolation polynomials with a limiter. But with the increasing of the order of interpolation polynomials, serious oscillation phenomena will appear, which will increase the computational error and affect instabilities of computations. So this thesis presents a kind of finite volume methods based on radial basis function (RBF). The RBF interpolation is a new kind of interpolation method. RBF interpolation has the following major advantages over polynomial approximation: more accurate if the same number of points used; very little changing of coefficients in the interpolation polynomial as the number of terms increase. RBF is a natural mesh-free method. The recent method is a high resolution approach. It works in a similar fashion as essentially non-oscillatory (ENO) finite volume scheme, and the difference is that it uses the radial basis function instead of high order interpolation polynomials on the adaptive stencils. Multiquadric (MQ) is one of the most commonly used in RBFs. It performs well for the interpolation of 2D scattered data. Therefore, we will concentrate on MQ RBF here. Firstly, we construct the new finite volume method based on radial basis function for scalar conservation laws. Then we generalize it to one and two dimensional problems of aerodynamics. Finally, the numerical experiments show that it is high accurate and efficient.