Linear hybrid-variable methods for advection equations

Abstract

We propose a general hybrid-variable (HV) framework to solve linear advection equations by utilizing both cell-average approximations and nodal approximations. The construction is carried out for 1D problems, where the spatial discretization for cell averages is obtained from the integral form of the governing equation whereas that for nodal values is constructed using hybrid-variable discrete differential operators (HV-DDO); explicit Runge-Kutta methods are employed for marching the solutions in time. We demonstrate the connection between the HV-DDO and Hermite interpolation polynomials, and show that it can be constructed to arbitrary order of accuracy. In particular, we derive explicit formula for the coefficients to achieve the optimal order of accuracy given any compact stencil of the HV-DDO. The superconvergence of the proposed HV methods is then proved: these methods have one-order higher spatial accuracy than the designed order of the HV-DDO; in contrast, for conventional methods that only utilize one type of variables, the two orders are the same. Hence, the proposed method can potentially achieve higher-order accuracy given the same computational cost, comparing to existing finite difference methods. We then prove the linear stability of sample HV methods with up to fifth-order accuracy in the case of Cauchy problems. Next, we demonstrate how the HV methods can be extended to 2D problems as well as nonlinear conservation laws with smooth solutions. The performance of the sample HV methods are assessed by extensive 1D and 2D benchmark tests of linear advection equations, the nonlinear Euler equations, and the nonlinear Buckely-Leverett equation.