In [L. Chen and R. Li, Journal of Scientific Computing, Vol. 68, pp.1172--1197, 2016], an integrated linear reconstruction was proposed for finitevolume methods. However, the geometric hypothesis of the mesh in thereconstruction is too restrictive to be satisfied by, for example, locallyrefined meshes or distorted meshes generated by arbitrary Lagrangian-Eulerianmethods in practical applications. In this paper, we propose an improvedintegrated linear reconstruction approach to get rid of the geometrichypothesis. The resulting optimization problem is a convex quadraticprogramming problem, and hence can be solved efficiently by classicalactive-set methods. The features of the improved integrated linearreconstruction include that i). the local maximum principle is fulfilled onarbitrary unstructured grids, ii). the reconstruction is parameter-free, andiii). the finite volume scheme is positivity-preserving when the reconstructionis generalized to Euler equations. A variety of numerical experiments arepresented to demonstrate the performance of the method.
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