  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.
