**backward_euler_fixed**,
a Python code which
solves one or more ordinary differential equations (ODE)
using the (implicit) backward Euler method, and a fixed point
iteration to solve the implicit equation.

Unless the right hand side of the ODE is linear in the dependent variable, each backward Euler step requires the solution of an implicit nonlinear equation. Such equations can be approximately solved using methods such as fixed point iteration, or an implicit equation solver like fsolve().

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

**backward_euler_fixed** is available in
a MATLAB version and
an Octave version and
a Python version.

backward_euler, a Python code which solves one or more ordinary differential equations (ODE) using the (implicit) backward Euler method, and fsolve() to solve the implicit equation.

euler, a Python code which solves one or more ordinary differential equations (ODE) using the forward Euler method.

leapfrog, a Python code which uses the leapfrog method to solve a second order ordinary differential equation (ODE) of the form y''=f(t,y).

midpoint, a Python code which solves one or more ordinary differential equations (ODE) using the (implicit) midpoint method.

midpoint_explicit, a Python code which solves one or more ordinary differential equations (ODE) using the (explicit) midpoint method, also called the modified Euler method.

midpoint_fixed, a Python code which solves one or more ordinary differential equations (ODE) using the (implicit) midpoint method, using a simple fixed-point iteration to solve the nonlinear equation.

rk4, a Python code which applies the fourth order Runge-Kutta (RK) algorithm to estimate the solution of an ordinary differential equation (ODE) at the next time step.

rkf45, a Python code which implements the Runge-Kutta-Fehlberg (rkf) solver for the solution of a system of ordinary differential equations (ODE).

trapezoidal, a Python code which solves one or more ordinary differential equations (ODE) using the (implicit) trapezoidal method.

trapezoidal_fixed, a Python code which solves one or more ordinary differential equations (ODE) using the (implicit) trapezoidal method, using a fixed point method to handle the implicit system.

- backward_euler_fixed.py, the source code.
- backward_euler_fixed.sh, runs all the tests.
- backward_euler_fixed.txt, the output file.