Simple Modifications of Monotonicity-Preserving Limiter

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