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

Abstract

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 very high-order accuracy which consists in specific restrictions considered in the least-squares minimization associated to the polynomial reconstruction. The method suffers from 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 we introduce the Reconstruction for Off-site Data (ROD) where polynomial reconstructions 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 very high-order accuracy and the optimal convergence rate is achieved.

Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries

“…The method consists in computing a local least‐squares approximation of function targeting some mesh elements, written in the general form as , where η is a vector of coefficients and is a basis function vector. Although any type of basis function vectors can be considered, polynomial basis functions present a high flexibility and are easy to construct . The local approximations are then used to compute accurate flux approximations and arbitrary high‐order accurate convergence can be achieved under mesh refinement.…”

“…The method introduced by Clain et al was later extended by Costa et al for curved domains with prescribed Dirichlet boundary conditions. As in the former, the method adapts the polynomial reconstructions associated to the boundary edges to enforce the boundary conditions at collocation points not belonging to the polygonal edges, successfully restoring the nominal convergence order of accuracy.…”

“…To avoid the use of curved mesh elements, a new technique proposed by Costa et al handles solely polygonal representative approximations of curved boundaries. As introduced in Section 3, for given boundary edge e i B , , B ∈ {D,N,R}, the prescribed boundary condition is evaluated at collocation point p i B with outward unit normal vector v i B .…”

“…The majority of the very high‐order accurate methods (more than the second‐order) are designed specifically for polygonal (or polyhedral) domains, and usually, numerical difficulties to obtain the nominal convergence order of accuracy arise when handling boundary conditions prescribed on curved boundaries. For a short literature review on the topic, the reader is referred to the introduction section in the work of Costa et al, which is summarized in the following. The classical approach to handle with curved boundary conditions is based on the isoparametric element method, which requires, on one side, the introduction of curved mesh elements, and on the other side, nonlinear transformations to map the local curved mesh elements onto the reference polygonal ones.…”

“…Costa et al introduced a new approach in the finite volume context, the reconstruction for off‐site data method (shortened to ROD method), which is capable of handling arbitrary smooth curved boundaries with a very high‐order convergence of accuracy. The strength and the novelty of the method rely on the fact that only polygonal mesh elements are used, even those generated to fit at best curved boundaries, for which a mismatch between the mesh boundary and the domain boundary exists.…”