Explanation results of RC simulation wih LTspice/Micro-cap
For LTSpice, you'll want to add an initial condition for the cap.
There are multiple ways to do this, the simplest is to modify the part value like so:



On a cap the ic set the initial voltage.  On an inductor is sets the initial current.  You can also specify node voltages etc.
ltwiki topic on initial conditions has more details.