# Linearized implicit time stepping

Consider the general FD implicit time stepping scheme

$\frac{x_{t+1} - x_t}{\Delta t} = f(x_{t+1})$,

where $x$ is the vector variable of interest and $f$ is some function, generally non-linear.

We can advance from $x_{t}$ to $x_{t+1}$ using Newton's method to solve the above non-linear equation. However, this can get quite costly.

If instead we write $f(x_{t+1}) \approx f(x_t) + J_f (x_t)(x_{t+1}-x_t)$, where $J_f$ denotes the Jacobian of $f$, we obtain the time-stepping scheme

$x_{t+1} - x_t = \left(I - \Delta t \, J_f(x_t) \right)^{-1} \, \Delta t f(x_t)$.

Can anybody tell me anything about this very general scheme, any references? What's its name? Stability properties?

For software, you can find the RODAS Fortran 77 implementation from Hairer and Wanner. I also implemented this family of methods in PETSc, which may be appropriate if you have a use for parallel linear algebra or sparse direct solvers. To experiment, you might start by looking at this ODE example written in Python which produces this figure. You can compare different methods using run-time options, e.g. -ts_type sundials will produce this figure.