Circuit Analysis With Non-Ideal Diodes
If you don't want the simplifications, you have to fall back to the general model of a diode:

$$
I = I_o \left( e^{\frac{eV}{nkT}}-1 \right)
$$

This equation relates the diode current to the diode voltage (it's V-I characteristic)


Io - is the diode reverse saturation current
k - Boltzmann's constant = 1.38e-23 Joules per Kelvin
T - Analysis temperature (Kelvin)
e - Magnitude of electric charge
n - Ideality factor (for silicon diodes, n=2 for small currents and approaches n=1 for large currents; in theory should always = 1)


You can now solve your circuit via the system of equations that it produces. Although you now have continuous V-I functions to describe your elements, a closed-form solution is not always guaranteed to exist. 

It is often necessary to use an iterative solution technique such as Newton-Raphson to approximate/approach the answer. This is what SPICE solvers do in the general case... and why they ask you for initial conditions (which can dramatically speed up the solution time).