This paper presents extended forward sensitivity analysis as a method to improve uncertainty quantification. By including the time step and potentially grid spacing as special sensitivity parameters, the forward sensitivity method is extended as one method to quantify numerical errors. Note that by integrating local truncation errors over the whole system through the forward sensitivity analysis process, the generated time step sensitivity information reflects global numerical errors. Discretization errors can be systematically compared against uncertainties due to other physical parameters. This extension makes the forward sensitivity method a much more powerful tool than other tools of its type to help uncertainty quantification. When the relative sensitivity of the time step to other physical parameters is known, the simulation is allowed to run at optimized time steps without affecting the confidence of the physical parameter sensitivity results. The time step forward sensitivity analysis method can also replace traditional time step convergence studies that are a key part of code verification, with much less computational cost. Two well-defined benchmark problems with manufactured solutions are utilized to demonstrate the method.