The Upwind Discontinuous Volume Method For The Convection-Diffusion Problem

In this paper,we firstly discuss the upwind cell-centered covolume method for the following convection-diffusion problem which uses'different rectangular grids called the primal and dual partitions.The con-centration is cell-centered,and the flux is co-cell centered.Chou et al.[48] established one-half order convergence in L2 norm.In this paper,optimal error estimates for both the scalar unknown and its flux are derived in the L2 norm,which improve the con-vergence results in [48].According to numerical examples,we confirm the correction of the convergence results.Then we consider an upwind discontinuous mixed covolume method on trian-gular grids for the above problem and establish one-half order convergence for the vector variable as well as the scalar variable in L2-norms.Furthermore,we obtain a first order convergence for the pure diffusion case.Since very few works have been done on the discontinuous mixed covolume method,it can be viewed as a new nu-merical method.Lastly,we introduce an upwind discontinuous finite volume method on triangu-lar grids for the above problem and derive a first order convergence for the scalar variable in the discrete H1-norms.Furthermore,we obtain a two order convergence for the scalar variable in the L2-norms.The upwind discontinuous mixed covolume method and the upwind discon-tinuous finite volume method combline the mixed covolume method and the finite volume method with the discontinuous Galerkin method,respectively,so they not only preserve the advantages of the mixed covolume method and the finite volume method,such as the simplicity of computation and maintaining the physical conser-vation,but also have the advantages of the discontinuous Galerkin method,such as parallelizability and localizability.Because of the upwind technique,they can elimi-nate some combination of nonphysical oscillation and excessive numerical dispersion.