An Integrated Quadratic Reconstruction for Finite Volume Schemes to Scalar Conservation Laws in Multiple Dimensions (CSIAM Transactions on Applied Mathematics, 2020)
We proposed a piecewise quadratic reconstruction method in multiple dimensions, which is in an integrated style, for finite volume schemes to scalar conservation laws. This integrated quadratic reconstruction is parameter-free and applicableon flexible grids. We show that the finite volume schemes with the new reconstructionsatisfy a local maximum principle with properly setup on time step length. Numerical examples are presented to show that the proposed scheme attains a third-orderaccuracy for smooth solutions in both 2D and 3D cases. It is indicated by numericalresults that the local maximum principle is helpful to prevent overshoots in numericalsolutions.
Keywords: Quadratic reconstruction; finite volume method; local maximum principle; scalar conservation law; unstructured mesh.