  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 a local maximum principle is too restrictive to be
satisfied by, for example, locally refined meshes or distorted meshes generated
by arbitrary Lagrangian-Eulerian methods in practical applications. In this
paper, we propose an improved integrated linear reconstruction approach to get
rid of the geometric hypothesis. The resulting optimization problem is a convex
quadratic programming problem, and hence can be solved efficiently by classical
active-set methods. The features of the improved integrated linear
reconstruction include that i). the local maximum principle is fulfilled on
arbitrary unstructured grids, ii). the reconstruction is parameter-free, and
iii). the finite volume scheme is positivity-preserving when the reconstruction
is generalized to the Euler equations. A variety of numerical experiments are
presented to demonstrate the performance of this method.
