Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids (Communications in Computational Physics, 2018)
In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp. 1172–1197, (2016)], an integrated linear reconstruction was proposed for finite volume methods on unstructured grids. However, the geometric hypothesis of the mesh to enforce alocal maximum principle is too restrictive to be satisfied by, for example, locally refinedmeshes or distorted meshes generated by arbitrary Lagrangian-Eulerian methods inpractical applications. In this paper, we propose an improved integrated linear reconstruction approach to get rid of the geometric hypothesis. The resulting optimizationproblem is a convex quadratic programming problem, and hence can be solved efficiently by classical active-set methods. The features of the improved integrated linearreconstruction include that i). the local maximum principle is fulfilled on arbitrary unstructured grids, ii). the reconstruction is parameter-free, and iii). the finite volumescheme is positivity-preserving when the reconstruction is generalized to the Eulerequations. A variety of numerical experiments are presented to demonstrate the performance of this method.