midpoint


midpoint, a C++ code which solves one or more ordinary differential equations (ODE) using the (implicit) midpoint method, with a version of fsolve() handling the associated nonlinear equation, and using a fixed time step.

Each midpoint step requires the solution of an implicit nonlinear equation representing a backward Euler step. A corresponding function has been devised, called fsolve_be(), which carries out the iterative solution process.

Licensing:

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

Languages:

midpoint is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a FreeFem++ version and a MATLAB version and an Octave version and a Python versionand an R version.

Related Data and codes:

midpoint_test

backward_euler, a C++ code which solves one or more ordinary differential equations (ODE) using the (implicit) backward Euler method, using a version of fsolve_be() for the implicit equation.

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

fsolve, a C++ code which solves systems of nonlinear equations, inspired by the fsolve() function in minpack(), with special interfaces fsolve_bdf2(), fsolve_be() and fsolve_tr() for handling systems associated with implicit ODE solvers of type bdf2, backward Euler, midpoint, or trapezoidal.

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

midpoint_fixed a C++ code which solves one or more ordinary differential equations (ODE) using the (implicit) midpoint method, using a fixed point method to solve the associated nonlinear equation.

midpoint_gsl_test, a C++ code which calls the Gnu Scientific Library (GSL) implicit midpoint method solver for ordinary differential equation (ODE), and uses gnuplot() to plot the resulting solution.

rk4, a C++ code which applies the fourth order Runge-Kutta (RK) algorithm to estimate the solution of an ordinary differential equation (ODE).

trapezoidal, a C++ code which solves one or more ordinary differential equations (ODE) using the (implicit) trapezoidal method, and a version of fsolve() to handle the nonlinear equation.

Reference:

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

Source Code:


Last revised on 10 November 2023.