High Order Finite Volume Hermite WENO Schemes For Convection Diffusion Equations

In recent years,the WENO(weighted essential non oscillatory)scheme has been greatly developed in computational fluid dynamics,This also promotes the development of other schemes based on this schemes.In this paper,a class of weighted essentially non oscillatory scheme based on Hermite polynomials called HWENO(Hermite WENO)scheme is proposed for solving convection dominated convection diffusion equations.It can achieve high-order accuracy in smooth region and keep strong discontinuity and no oscillation at discontinuity.In this paper,the calculation of diffusion term is introduced into the convection term.First,the equation is written as a conservative form,Then it is discretized by HWENO scheme.The constructed spatial term was based on the high order accuracy Hermite interpolation,finite volume formulation,and the time term was advanced by using the nonlinearly stable Runge-Kutta method.In the two-dimensional case,because it involves the reconstruction of mixed derivative terms,it is difficult to use HWENO to calculate the solution.Therefore,in order to keep the compactness of the result,the mixed derivative term in two-dimensional case is analyzed by Hermite interpolation of three points.Although the construction process of HWENO is based on the intrinsic WENO,the difference between them is that HWENO reconstruction needs not only the original function but also the first derivative of the function,while the WENO scheme only uses the original function to participate in the reconstruction.Because of the use of more conditions,HWENO scheme can be a more compact scheme than the WENO scheme.For example,for a fifth-order WENO reconstruction,five nodes are needed;But for the HWENO reconstruction of the fifth order,because of the use of the first derivative of the function at the node,only three points are needed.Finally,A lot of numerical examples show that the scheme is suitable for deal with this class of equations.