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We proposed a piecewise quadratic reconstruction method in multipledimensions, which is in an integrated style, for finite volume schemes toscalar conservation laws. This integrated quadratic reconstruction isparameter-free and applicable on flexible grids. We show that the finite volumeschemes with the new reconstruction satisfy a local maximum principle withproperly setup on time steplength. Numerical examples are presented to showthat the proposed scheme attains a third-order accuracy for smooth solutions inboth 2D and 3D cases. It is indicated by numerical results that the localmaximum principle is helpful to prevent overshoots in numerical solutions.

An Integrated Quadratic Reconstruction for Finite Volume Schemes to Scalar Conservation Laws in Multiple Dimensions