We present simple modifications of standard monotonicity-preserving limiter
s that provide either sign preservation or alternative bounding values for
the resulting numerical schemes (e.g., that the solution remain between zer
o and one rather than preserving monotonicity). These limiters can be easil
y implemented in Godunov-type methods by modifying the reconstruction step
of the algorithm. These modifications allow methods to achieve greater form
al accuracy, for example by improving the first-order accuracy in the L-inf
inity norm. The greatest advantage of our approach is its natural extension
to more than one spatial dimension, and to systems of equations without op
erator splitting.