Integrated Linear Reconstruction for Finite Volume Scheme on Arbitrary Unstructured Grids
This work addresses a key limitation in computational fluid dynamics by enabling robust finite volume schemes on complex, distorted, or locally refined meshes without geometric constraints.
The authors propose an improved integrated linear reconstruction for finite volume schemes on arbitrary unstructured grids that eliminates restrictive geometric hypotheses, ensuring the local maximum principle holds on any mesh. The method is parameter-free, positivity-preserving for Euler equations, and validated through numerical experiments.
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.