# midpoint

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

Unless the right hand side of the ODE is linear in the dependent variable, each midpoint step requires the solution of an implicit nonlinear equation. This code uses the nonlinear equation solver fsolve().

### Languages:

midpoint is available in a FreeFem++ version and a MATLAB version and an Octave version and a Python version and an R version.

### Related Data and Programs:

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

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_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.

### Reference:

1. Catalin Trenchea, John Burkardt,
Refactorization of the midpoint rule,
Applied Mathematics Letters,
Volume 107, September 2020.

### Source Code:

Last revised on 07 April 2021.