How to model Nernst equation for numeric solution?

I want to write a program (this is not programming question!!) that numerically solves voltage across virtual battery-resistor circuit over time. But sometimes explanations is too overcomplicated, not using simple terms. This approach favors short length, but slowers understanding alot (requires years practicing math, to get clear vision of "what is going on", just seeing math notation). Let me please explain on example that I can understand.

For a RC cirtuit, its very easy. When solving ODE numerically you have some bunch of changing values, measured at fixed snapshots of time t. To know voltage you may do V(t) = V(t-1) + dV(t-1)*dT. That is a main calculus loop of a solution program. You know V(t-1), dT is fixed. And dV(t) = -V(t)/RC. So far so good, this resource is a bad explanation example. Because solver does not require exp, it arises from solution. It does not require integral, so on.. No simple formula dv/dt = .... At least they have graph (reversed, for charging) from where you may get simple intuition:

Now I want to simulate chemical battery (accumulator), google said I need Nernst equation. But wiki guys again start with Ox, arrows ->, kind of some other fancy symbols, that should occasionally get to Red?.. Oh, I was just about writing a sim program within 5 minutes, but they dont get answer. All that I want is "to simulate it in digital ODE solver do this equation, where parameters are..". So, finally, my question is – can you help me to rephrase in simple words? Or how to calculate dV(t)=? for a battery?

PS. A battery starts at t=0, with known V(0) = V0, conneted to resistance R again, so on.. and i want kind of simple model that would draw me something like