A general approach to enhance slope limiters on non-uniform rectilinear grids

TL;DR

This paper develops a general method to improve slope limiters in high-resolution finite volume methods, ensuring TVD stability, second-order accuracy, and symmetry preservation on highly non-uniform rectilinear grids.

Contribution

It introduces a novel approach extending classical procedures to non-uniform grids, providing conditions for slope limiters to maintain stability and accuracy.

Findings

Enhanced limiters satisfy TVD and second-order accuracy conditions.

Numerical tests confirm improved stability and accuracy on non-uniform grids.

Classical limiters are insufficient for highly non-uniform grids without modification.

Abstract

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 non-uniform rectilinear grids. However, this strategy may lead to either loss of total variation diminishing (TVD) stability for 1D linear problems or the loss of formal second-order accuracy if the grid is highly non-uniform. This is especially true when the limiter function is not piecewise linear. Numerical evidences are provided to support this argument for two popular finite volume strategies: MUSCL 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 non-uniform 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 non-uniform rectilinear grids.