There is following problem:
$$C_{(q)}\frac{\partial v}{\partial t}= i_{(t)}$$
a differential equation for current through capacitor, where $i_{(t)}$ denotes a rest of the circuit including serial and parallel resistance (and other non-linear components, not changing the nature of the problem). Now, discretizing the problem with 1st order method:
$$\frac{v_{[k+1]} - v_{[k]}}{\Delta t} = \frac{1}{C_{(q)}}*i_{(k)}$$
As the the notation $C_{(q)}$ shows, the $C$ is a function of free parameter $q$, such that $C_{(q)} = 0$ for $q_0$. This poses an immediate question, on how to solve case for $q_{0}$ when $\frac{1}{C_{(q_0)}} \to\infty$.
From physical point of view, the $C=0$ is valid case (e.g. the value of C is lowered/neglected, so the circuit behaves statically), but solution based on differentiation of $v$ fails, because the $C_{(q_0)}$ is in denominator.
QUESTIONs
- Can the problem be solved by integration of $i$, instead by derivation of $v$ ?
- Does (any) numerical integration method, have the same/similar properties in terms of errors, as the Runge-Kutta methods (max error is kept within known limits, also higher order method, lower error).
P.S. The problem is extremely simplified, the term $i(t)$ is complex function for whole circuit. Please dont post analytical methods for solving differential equations !
EDIT: More info/insight view into the problem.
The following holds and should be taken into consideration (source):
$$\frac{\partial C_{(q)}}{\partial t}v + C_{(q)}\frac{\partial v}{\partial t}= i_{(t)}$$