On supra-convergence of the finite volume method for the linear advection problem.

*(English)*Zbl 1388.76186Summary: This paper investigates the supra-convergence phenomenon that one can observe in the upwind finite volume method for solving linear convection problem on a bounded domain but also in finite difference scheme with non-uniform grids. Although the scheme is no longer consistent in the finite difference sense and Lax-Richtmyer theorem not suitable, it is a well-known convergent method. In order to analyze the convergence rate, we introduce what we call a geometric corrector, which is associated with every finite volume mesh and every constant convection vector. Under a local quasi-uniformity condition and if the continuous solution is regular enough, there is a link between the convergence of the finite volume scheme and this geometric corrector : the study of this latter leads to the proof of the optimal order of convergence. We then focus our attention on an uniformly refined mesh of quadrangles and on a series of independent meshes of triangles and tetrahedrons. In these latter cases, a loss of accuracy is observed if there exists in the family of meshes a fixed straight line parallel to the convection direction.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |