Why do we get oscillations in Euler's method of integration and what i

[ErezAgh]

When using Euler's method of integration, applied on a stochastic differential eq. :

For example - given
d/dt v=−γvΔt+sqrt(ϵ⋅Δt)Γ(t)
we loop over

v[n+1]=v[n]−γv[n]Δt+sqrt(ϵ⋅Δt)Γn.
(where −γv[n] is a force term, can be any force and Γn is some gaussian distributed random variable. ) .

Then if we choose Δt not small enough, we eventually get (if we run over long times, meaning many repeated iterations) that the solutions become "unstable and oscillations appear around the analytic solution, with amplitude becoming larger and larger with time" (~ collected from many different sources I found on the internet that mention the problem but don't discuss it in depth).

Why are these actual OSCILLATIONS, and not simply random fluctuations? What is the period of these oscillations?

 

[pbuk]

Why are these actual OSCILLATIONS, and not simply random fluctuations?

For the same reason single-step numerical integration is unstable for non-stochastic problems when the step size is large enough so that each step 'overshoots' the analytic solution.

What is the period of these oscillations?

Er, the step size.

 

[pasmith]

When using Euler's method of integration, applied on a stochastic differential eq. :

For example - given
d/dt v=−γvΔt+sqrt(ϵ⋅Δt)Γ(t)
we loop over

v[n+1]=v[n]−γv[n]Δt+sqrt(ϵ⋅Δt)Γn.
(where −γv[n] is a force term, can be any force and Γn is some gaussian distributed random variable. ) .

Then if we choose Δt not small enough, we eventually get (if we run over long times, meaning many repeated iterations) that the solutions become "unstable and oscillations appear around the analytic solution, with amplitude becoming larger and larger with time" (~ collected from many different sources I found on the internet that mention the problem but don't discuss it in depth).

Why are these actual OSCILLATIONS, and not simply random fluctuations?

Consider the non-stochastic ODE $$y' = -ky$$ with $k > 0$ and $y(0) = 1$. This can be solved analytically: $$y(t) = e^{-kt}.$$ Note that $y \to 0$ as $t \to \infty$. Applying Euler's method with step size $h > 0$ we obtain $$y_{n+1} = y_n - hky_n = y_n(1 - hk)$$ which again can be solved analytically:
$$y_n = (1 - hk)^n.$$ Thus the error at time $t = nh$ is given by $$\epsilon_n = y(nh) - y_n = e^{-nhk} - (1 - hk)^n.$$ If $hk > 2$ then $1 - hk < -1$, so that $$\epsilon_n = e^{-nhk} - (-1)^n|1 - hk|^n.$$ Thus, since $|1 - hk| > 1$ and $e^{-nhk} < 1$, we see that $|\epsilon_n| \to \infty$ and that $\epsilon_n$ is alternatively positive and negative (which is what "oscillates" means in this context).

 

[ErezAgh]

Consider the non-stochastic ODE $$y' = -ky$$ with $k > 0$ and $y(0) = 1$. This can be solved analytically: $$y(t) = e^{-kt}.$$ Note that $y \to 0$ as $t \to \infty$. Applying Euler's method with step size $h > 0$ we obtain $$y_{n+1} = y_n - hky_n = y_n(1 - hk)$$ which again can be solved analytically:
$$y_n = (1 - hk)^n.$$ Thus the error at time $t = nh$ is given by $$\epsilon_n = y(nh) - y_n = e^{-nhk} - (1 - hk)^n.$$ If $hk > 2$ then $1 - hk < -1$, so that $$\epsilon_n = e^{-nhk} - (-1)^n|1 - hk|^n.$$ Thus, since $|1 - hk| > 1$ and $e^{-nhk} < 1$, we see that $|\epsilon_n| \to \infty$ and that $\epsilon_n$ is alternatively positive and negative (which is what "oscillates" means in this context).

Thanks a lot both!

1) I see why it oscillates with a frequency of 1 step, (-1)^n, so this means you normalized the step size to be Er=1?
2) Silly question perhaps, but can this be written in the form of a sine() than?

 

[CellCoree]

The oscillations in Euler's method of integration are a result of the approximation used in the numerical algorithm. In this method, the differential equation is approximated by a series of smaller steps, with the assumption that the solution remains constant within each step. However, in reality, the solution may change significantly within each step, leading to errors in the numerical approximation.

As a result, these errors accumulate over time, causing the solution to deviate from the true solution and exhibit oscillations. These oscillations are not random fluctuations, but rather a result of the cumulative error in the numerical approximation.

The period of these oscillations can vary depending on the specific problem and the chosen step size. In general, smaller step sizes will lead to shorter periods of oscillations, as the errors are smaller and accumulate at a slower rate. However, there is no fixed period for these oscillations, as they are a result of the specific dynamics of the system being modeled.

To avoid these oscillations, it is important to choose a small enough step size and to carefully consider the dynamics of the system being modeled. Other methods of integration, such as the Runge-Kutta method, may also be more accurate and stable for certain types of differential equations.