A very high-order finite volume method for the one dimensional convection diffusion problem

Stéphane Clain (1, 2) , Gaspar Machado (2) , Rui M. S. Pereira (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 DMA - Departamento de Matemática e Aplicaçoes
3 DMA - Departamento de Matemática e Aplicações

A new finite volume method for one dimensional convection diffusion problems is designed. The scheme is based on the Polynomial Reconstruction Operator (PRO-scheme) where a 5-degree polynomial representation is provided for each cell using the mean-values of the neighboring cells. Convective and diffusive numerical fluxes derive from the polynomial reconstruction and lead to a finite volume scheme of sixth-order of accuracy. Numerical examples are proposed to show the effectiveness of the method and the ability to manage the convection and the diffusion independently, {\it i.e.} the scheme is still functional even with vanishing viscosity.

Mots clés: Polynomial Reconstruction Operator (PRO); Finite Volume; very high-order; convection-diffusion; Polynomial Reconstruction Operator (PRO).

Domaines: Analyse numérique [math.NA]

Stéphane Clain, Gaspar Machado, Rui M. S. Pereira. A very high-order finite volume method for the one dimensional convection diffusion problem. 2011. ⟨hal-00653968⟩