Handling Time Terms in Implicit Difference Method


I am encountering some issues related to very small time steps in my numerical simulations and would appreciate your assistance.

In my model, the time step dt is extremely small, approximately on the order of 10^-12 . I am using the implicit difference method to handle the time derivative, and there are two ways to approach this:

Using dt as a divisor (i.e., dividing the change in the variable by dt).
Using dt as a multiplier (i.e., multiplying other terms by dt so that the time difference term eliminates dt).
I have a concern that using dt as a multiplier will result in very small values in the matrix, which might affect the solution process. This is my own observation, and I am not sure if it is correct.

Most of the examples I have seen use dt as a multiplier. However, given the very small time step in my model, I am worried that this approach may impact the computational accuracy and stability.

Could you please advise which method is recommended in this situation and why?

Thank you very much for your help!