A high-order interpolation for the finite volume method: The Coupled Least Squares reconstruction

Abstract

We consider the finite volume method on irregular grids for conservation laws with a particular polynomial reconstruction. This reconstruction is based on a least squares approach to compute consistent approximations of the first, second and third order derivatives. The resulting reconstruction is cubic. It is called the Coupled Least Squares reconstruction. It is obtained by a three stages iteration. At each iteration, only data located in a compact stencil, and not beyond, are accessed. A linear stability analysis is given in the case of regular and irregular one-dimensional grids. Numerical results for various problems, including shock tubes, vortex and acoustic wave propagation, support the interest of this approach. The reconstruction algorithm is presented in detail in the one dimensional case. An outline of the multidimensional case in also given.This work was announced in Haider et al. (2014) [14].