Geodesic
MUSCL-scheme of the third order of accuracy on a non-uniform structured grid
Numerical methods and programming, Tome 24 (2023) no. 4, pp. 386-407.

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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.
Keywords: MUSCL-scheme; high-order reconstruction; non-uniform mesh; structured mesh; finite volume method; Navier-Stokes equations. 
@article{VMP_2023_24_4_a3,
     author = {A. R. Kocharina and D. V. Chirkov},
     title = {MUSCL-scheme of the third order of accuracy on a non-uniform structured grid},
     journal = {Numerical methods and programming},
     pages = {386--407},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2023_24_4_a3/}
}
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A. R. Kocharina; D. V. Chirkov. MUSCL-scheme of the third order of accuracy on a non-uniform structured grid. Numerical methods and programming, Tome 24 (2023) no. 4, pp. 386-407. http://geodesic.mathdoc.fr/item/VMP_2023_24_4_a3/