I am working on a project that uses numerical integration to solve some differential equations. Not coming from a solid math background, I have a problem understanding some integration methods. Specifically, I have the following equation:

$$ \frac{d y}{d t} = \frac{u - y}{v} \,, $$ where $u$ and $v$ are constants and $y(0)$ is known.

I solved this initially with a simple Euler integration and then with some other higher-order methods (Runge-Kutta 4). Then, I found another numerical integrator that did: $$ y_{n+1} = y_{n} + (u - y_{n})(1 - e^{\frac{-\Delta t}{v}}) \,. $$

My question is: what is this integration method? I checked and the results are basically the same, but I had no idea about this technique and I would like to know more about it.


Its just the analytical answer to your equation.

$y(t) = c_1 e^{-\frac{t}{v}}+u$, with an initial condition $y(0) = c_1 + u = y_0$ which gives you $y(t) = (y_0-u) e^{-\frac t v}+u = y_0 + (u-y_0)(1-e^{-\frac t v})$

Just call $y_{n+1} = y(t)$ and $y_n = y_0$ with $\Delta t = t$ and you obtain your equation. This is not a general numerical method which applies to other differential equations.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.