A numerical simulation of selective catalytic reduction (SCR) for NO with $NH_3$ is conducted over the $V_2O_5/TiO_2$ and $WO_3-V_2O_5/TiO_2$ catalysts. The governing $NH_3$ and NO transport equations are considered by using the time-dependent FCT (Flux-Corrected Transport) algorithm. After a validating simulation for $NH_3$ step feed and shut-off experiments is analyzed, transient behavior of $NH_3$ and NO concentration in a SCR catalyst is investigated by changing such parameters as inflow $NH_3$ concentration, temperature of the catalyst, and $NH_3$/NOx ratios.