Improved total variation diminishing schemes for advection simulation on arbitrary grids

The requirements for flux limiter functions preserving total variation diminishing (TVD) are derived based on a 1D nonuniform grid, and a new TVD region is determined to fit arbitrary 1D grids. Some second‐order TVD schemes called improved TVD schemes are developed, such as modified Van Leer scheme, modified Van Albada scheme, and modified SUPERBEE scheme. Then, they are extended to 2D grids. Because the element sizes and face positions are taken into account, good behaviors are observed in the implementations in both 1D and 2D cases for pure advection simulation. That is, good conservation, better monotonicity, and higher accuracy are maintained by the improved TVD schemes compared with the present ones deduced from uniform grids, and they keep superiorities even when implemented on poor grids. Among all the improved TVD schemes, the modified SUPERBEE is only recommended for poor 1D grids, but the modified Van Leer scheme can suit both poor 1D and 2D grids. Copyright © 2011 John Wiley & Sons, Ltd.