How do spice simulators work

Is anyone aware of steps that are involved in the calculation of the waveform for a simple ramp or input to a transmission line and possibly a capacitor in between.

I know I can simulate the waveform using hspice or any simulator. Now I want to derive my own equations to arrive at the conclusions of seeing the waveform at the output. Any pointer will be helpful.

It all boils down to derive the differential equation of your system. If you have one input and one output, it should be a single equation. If you are able to solve it, you have the exact result, and not the approximate result that spice gives you.

So, if I have a transmission line and I split it in LC form, create a differential equation, and solve it ( let us say, using numerical methods), then I get the output curves....correct ?

Not exactly. Here is what you'll see if you model lossless line with LCs: a lot of oscillations, quite different from ideal line response. The better way to model such a line is to consider V and I inside the line as a combination of two waves: forward and reflected. I think you can find detailed description in the Internet. Very short: for each wave V=Iz, where z is characteristic impedance of the line. The wave's voltages sum inside the line, currents subtract, and both should satisfy boundary conditions (at the ends of the line).
I don't know what methods can be used for lossy line, other than modeling it directly by RL/CG chains.
Thank you.