SUMMARY 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.

SUMMARYFor advection schemes with flux limiters derived on one‐dimensional grids to two‐dimensional (2D) unstructured triangular ones. In this method, the variables located normal to this face are taken into more account to compute the flux, by means of defining required nodes along the line through the center point of the considered face and perpendicular to it. Besides, the new method adopts the improved total variation diminishing schemes in, which consider the face position between the two neighboring cells as well as the size differences of the related cells; therefore, it fits 2D unstructured grids well. Compared with the present flux limiting methods for 2D unstructured grids, a higher accuracy and efficiency and a good monotonicity are achieved by the new method.Copyright © 2012 John Wiley & Sons, Ltd.

Aiming at improving the computational eciency without accuracy losses for surface water ow simulation, this paper presents a structured but non-uniform grid system incorporated into a Godunov-type nite volume scheme. The proposed grid system can detect the key topographic features in the computational domain where high-resolution mesh is in need for reliably solving the shallow water equations. The mesh renement is automatically carried out in these areas while the mesh in the rest of the domain remains coarse. The criterion determining the renement is suggested by a dimensionless number with a xed value of 0.2 after sensitivity analysis. Three laboratory and eld-scale test cases are employed to demonstrate the performance of the model for ow simulations on the new non-uniform grids. In all of the tests, the grid system is shown to successfully generate high-resolution mesh only in those areas with abruptly changing topographic features that dominate the ooding processes. To produce numerical solutions of similar accuracy, the non-uniform grid based model is able to accelerate by about two times comparing with the ne uniform grid based counterpart.

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