A framework to perform high‐order deconvolution for finite‐volume method on simplicial meshes

Abstract

In this article, a new framework to design high-order approximations in the context of node-centered finite volumes on simplicial meshes is proposed. The major novelty of this method is that it relies on very simple and compact differential operators, which is a critical point to achieve good performances in the High-performance computing context. This method is based on deconvolution between nodal and volume-average values, which can be conducted to any order. The interest of the new method is illustrated through three different applications: mesh-to-mesh interpolation, levelset curvature computation, and numerical scheme for convection. Higher order can also be achieved within the present framework by introducing high-rank tensors. Although these tensors feature much symmetries, their manipulation can quickly become an overwhelming task. For this reason and without loss of generality, the present articles are limited to third-order expansion. This method, although tightly connected to the k-exact schemes theory, does not rely on successive corrections: the high-order property is obtained in a single operation, which makes them more attractive in terms of performances.

Numerical simulation of boiling on unstructured grids

“…In [9], the authors proposed a new framework to reconstruct differential operators up to a chosen accuracy order within the node-centered finite-volume method on unstructured simplicial grids, in two and three dimensions. This method has been implemented in the Boiling solver and is detailed below.…”

“…This step is called deconvolution. In [9], the authors propose a third-order polynomial reconstruction of Z I from Z Ω I , together with a second-order polynomial reconstruction of ∇Z I and a first-order polynomial reconstruction of ∇∇Z I .…”

“…(6.34) to the invertibility of matrix C O3 I . This matrix is invertible iif node x I has a sufficient number of direct neighbor nodes (see [9] for details). In case of an insufficient number of direct neighbors to perform a third-order reconstruction, a lower-order reconstruction can be applied, or the third-order reconstruction performed on some direct neighbor node can be used to reconstruct Z at order three on the initial node.…”

“…(6.57) and (6.58) can also be weighted by the relative distance θ x I ,x J in order to improve accuracy. In [9], the interface curvature computed using Eq. (6.56) is shown to be second-order accurate on cartesian grids and first-order accurate on unstructured grids.…”

A massively parallel accurate conservative level set algorithm for simulating turbulent atomization on adaptive unstructured grids

“…It is useful to define the unit pair direction e ik = ∆x ik /∆x ik , which will be used in the following sections. More details about the high-order finite-volume framework used can be found in [41].…”