MUSCL-scheme of the third order of accuracy on a non-uniform structured grid

An upwind finite volume scheme with third-order MUSCL-reconstruction at the cell interface is extended to non-uniform structured grids. The order of approximation of the original MUSCL-scheme with reconstruction using constant coefficients and the modified MUSCL-scheme with coefficients dependent on the grid steps is investigated for 1D nonlinear transport equation. It is shown that the order of approximation depends on the type of non-uniform grid. The cases of a grid with a constant clustering law and an arbitrary non-uniform grid are considered. It is shown analytically and numerically, that the non-uniform MUSCL-scheme with coefficients depending on the grid spacing has the third order of approximation on a non-uniform grid with a constant clustering law and the second order on an arbitrary grid. It is also shown that the MUSCL-scheme with constant coefficients does not approximate the original equation at all on an arbitrary non-uniform grid. Non-uniform MUSCL-reconstruction is introduced into the numerical algorithm for calculating incompressible fluid flows. Higher accuracy of the proposed scheme is demonstrated for a 2D problem of the flow around a circular cylinder and for a 3D fluid flow in a hydraulic turbine.