Suppose we're looking at a first order ODE of the form $$ \frac{dx}{dt}=-\lambda x+ b u $$ where $\lambda$ and b are functions of $x$ and $u$ is an 'energy generating' term which is a function of $x$ and $t$ (my use of the equation is as an energy conservation/heat exchange equation with $x$ being the temperature and $u$ the internal heat generation). There is a method that uses a laplace transform making it $$ X(s)=\frac{x_{0}}{s-\lambda}+\frac{bu}{s(s-\lambda)} $$ where $x_{0}=x(t_{0})$ and then a reverse laplace transforms to bring it to this next form: $$ x(t)=e^{-\lambda t} x_{0} + (1-e^{-\lambda t})bu/\lambda $$ Then, assuming $\lambda$, $b$ and $u$ to be constant in the time interval $[t,t+\Delta t]$ we derive the next finite difference rule: $$ x_{n+1}=e^{-\lambda_{n} \Delta t} x_{n} + (1-e^{-\lambda_{n} \Delta t})b_{n}u_{n}/\lambda_{n} $$ This method is supposed to aid in convergence to the steady state solution even when using very large time steps.
Since I don't know the name of this method I haven't been able to find any information regarding it. I would like to find more information about this method and how to use it on very similar but slightly different forms ODE's.
Does anyone know the name of this method and can direct me to relevant sources?
I found the name of the method (finally), so for future inquiries: It's a class of numerical methods for solution of differential equations called
Exponential Integrator. As usual, Wikipedia has an article about it, so that's a good place to start.Cheers