A GENERAL APPROACH TO ENHANCE SLOPE LIMITERS IN MUSCL SCHEMES ON NONUNIFORM RECTILINEAR GRIDS

Most slope limiter functions in high-resolution finite volume methods to solve hyperbolic conservation laws are designed assuming one-dimensional uniform grids, and they are also used to compute slope limiters in computations on nonuniform rectilinear grids. However, this strategy may lead to either loss of the total variation diminishing (TVD) stability for one-dimensional linear problems or loss of formal second-order accuracy if the grid is highly nonuniform. This is especially true when the limiter function is not piecewise linear. Numerical evidence is provided to support this argument for two popular finite volume strategies: monotonic upstream-centered scheme for conservation laws in space and method of lines in time (MUSCL-MOL) and capacity-form differencing. In order to deal with this issue, this paper presents a general approach to study and enhance the slope limiter functions for highly nonuniform grids in the MUSCL-MOL framework. This approach extends the classical reconstruct-evolve-project procedure to general grids, and it gives sufficient conditions for a slope limiter function leading to a TVD stable, formal second-order accuracy in space, and symmetry-preserving numerical scheme on arbitrary grids. Several widely used limiter functions, including the smooth ones by van Leer and van Albada, are enhanced to satisfy these conditions. These properties are confirmed by solving various one-dimensional and two-dimensional benchmark problems using the enhanced limiters on highly nonuniform rectilinear grids.