High-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection-diffusion equation with Dirichlet condition

Ricardo Costa (1) , Stéphane Clain (2) , Raphaël Loubère (1, 3) , Gaspar J Machado (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 CMAT - Centro de Matemática [Minho]
3 CNRS - Centre National de la Recherche Scientifique

Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers of several drawbacks, particularly, the use of curved elements that requires sophisticated meshing algorithms. We propose a new method where the physical domain and the computational domain are distinct and introduce the Reconstruction of Off-site Data (ROD) where polynomial reconstruction are carried out on the mesh using data localized outside of the computational domain, namely the Dirichlet condition situated on the physical domain. A series of numerical tests assess the accuracy, convergence rates, robustness, and efficiency of the new method and show that the boundary condition is fully integrated in the scheme with a high-order accuracy and the optimal convergence rate is achieved.

Mots clés: High-order finite volume method; curved boundaries; Reconstruction of Off-site Data (ROD)

Domaines: Analyse numérique [math.NA]

Ricardo Costa, Stéphane Clain, Raphaël Loubère, Gaspar J Machado. High-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection-diffusion equation with Dirichlet condition. Applied Mathematical Modelling, 2018, 54, ⟨10.1016/j.apm.2017.10.016⟩. ⟨hal-01399426⟩