High-order mass conservative compact characteristic finite volume methods for advection-dominated diffusion equations

Abstract

In this work, high-order mass conservative compact characteristic finite volume methods are derived for advection-dominated diffusion equations. To overcome the restriction of the time step and obtain higher temporal accuracy, the method of characteristics is employed to treat the advection term. The compact finite volume technique is used to construct spatial fourth-order schemes with fewer blocks than traditional methods. Moreover, the Alternating Direction Implicit (ADI) method is utilized to effectively solve two-dimensional problems. The proposed compact finite volume methods can achieve temporally second-order and spatially fourth-order accuracy and possess mass conservation by implementing conservative interpolations and averaging along the characteristics. The temporal and spatial accuracy of the schemes, as well as the mass conservation property, are demonstrated in numerical tests on the solution of Gaussian pulse moving and viscous Burgers' equations. The computational results of transporting a square and a slotted cylinder show the good performance of the proposed methods in preserving mass and simulating the transport of concentration distributions with steep gradients.