This solver solves an explicit ordinary differential equation using the implicit Euler method.

We want to approximate the solution to the ordinary differential equation of the form,

\[\dot{y} = f(t,y), \quad y(t_0) = y_0 .\]

Using the implicit Euler method, the approximation is defined as follow,

\[y_{n+1} = y_n + hf(t_{n+1},y_{n+1})\]

with \(h\) being the step-size and \(y_n\) the previous solution to the equation.

- State events (root funtions) : True
- Step events (completed step) : True
- Time events : True

Import the solver together with the correct problem:

```
from assimulo.solvers import ImplicitEuler
from assimulo.problem import Explicit_Problem
```

Define the problem, such as:

```
def rhs(t, y): #Note that y are a 1-D numpy array.
yd = -1.0
return N.array([yd]) #Note that the return must be numpy array, NOT a scalar.
y0 = [1.0]
t0 = 1.0
```

Create a problem instance:

```
mod = Explicit_Problem(rhs, y0, t0)
```

Note

For complex problems, it is recommended to check the available examples and the documentation in the problem class, `Explicit_Problem`

. It is also recommended to define your problem as a subclass of `Explicit_Problem`

.

Warning

When subclassing from a problem class, the function for calculating the right-hand-side (for ODEs) must be named *rhs* and in the case with a residual function (for DAEs) it must be named *res*.

Create a solver instance:

```
sim = ImplicitEuler(mod)
```

Modify (optionally) the solver parameters.

Parameters:

`atol`

Defines the absolute tolerance(s) that is to be used by the solver.`backward`

Specifies if the simulation is done in reverse time.`clock_step`

Specifies if the elapsed time of an integrator step should be timed or not.`display_progress`

This option actives output during the integration in terms of that the current integration is periodically printed to the stdout.`h`

Defines the step-size that is to be used by the solver.`newt`

Maximal number of Newton iterations.`num_threads`

This options specifies the number of threads to be used for those solvers that supports it.`report_continuously`

This options specifies if the solver should report the solution continuously after steps.`rtol`

Defines the relative tolerance that is to be used by the solver.`store_event_points`

This options specifies if the solver should save additional points at the events, \(t_e^-, t_e^+\).`time_limit`

This option can be used to limit the time of an integration.`usejac`

This sets the option to use the user defined jacobian.`verbosity`

This determines the level of the output.

Simulate the problem:

Information:

`ImplicitEuler.get_options()`

Returns the current solver options.`ImplicitEuler.get_supports()`

Returns the functionality which the solver supports.`ImplicitEuler.get_statistics()`

Returns the run-time statistics (if any).`ImplicitEuler.get_event_data()`

Returns the event information (if any).`ImplicitEuler.print_event_data()`

Prints the event information (if any).`ImplicitEuler.print_statistics()`

Prints the run-time statistics for the problem.