I want to see the effect of parameter uncertainty in the Euler method for ODEs.
For a differential equation:
$dx/dt=f$
with initial condition $x(0)=xo$ and a function $f$ (that has uncertain parameters and could be a nonlinear function of $x$), the Euler method is:
$x_{n+1}=x_{n} + f_n.\Delta t$
where $n$ is the step index of the integration. The variance for this step is:
$ \sigma^2_{xn+1}=\sigma^2_{xn}+\Delta t^2.\sigma^2_{fn}+2 \Delta t. \sigma_{xn,fn} $
Since $x_n$ and $f_n$ can be strongly correlated, I want to know how to estimate $\sigma_{xn}$ and $\sigma_{xn,fn}$ for the general case.
For example, let's consider:
$f=-k*x$
where $k$ is normally distributed with mean $k_m$ and variance $\sigma^2_k$. Then:
$ \sigma^2_{xn+1}=\sigma^2_{xn}+(x_n.k_m)^2[\sigma^2_{xn}/x^2_n+\sigma^2_{k}/k_m^2+2.\sigma_{xn,-k}/(-x_n.k_m)].\Delta t^2+2 \Delta t. \sigma_{xn,-xn.k} $
In this case $x_n$ and $k$ become very correlated after the first step. How can I calculate $\sigma_{xn.-k}$ and $\sigma_{xn,-xn.k}$ ?
Note: I know that this problem can be solved analytically or using Monte Carlo for the complete integration (sample k, solve the ODE, repeat for different values of k). This is not what I am interested in. Also, I know that rearranging $x_{n+1}=x_n.(1-k\Delta t)$ makes the problem easier, but I want a formulation similar to the more general case.
