conservation

Location: http://people.sc.fsu.edu/~jburkardt/classes/math1090_2020/conservation/conservation.html

conservation looks at how certain physical laws require that some quantities be conserved over time; and how some ODE solvers can satisfy such a requirement.

The notes:

• conservation.pdf
• conservation.tex

Matlab:

• euler.m, a revised version of the forward Euler code, which can handle systems of ODEs.
• kepler_conserved.m, evaluates the conserved quantity for the Kepler ODE.
• kepler_deriv.m, defines the derivatives associated with the Kepler ODE.
• kepler_euler.m, solves the Kepler ODE using euler().
• kepler_midpoint.m, solves the Kepler ODE using midpoint_fixed().
• kepler_ode_test.m, runs all the Kepler ODE solvers.
• kepler_ode45.m, solves the Kepler ODE using ode45().
• kepler_ode_test.m, calls kepler_euler, kepler_midpoint, kepler_ode45.
• midpoint_fixed.m, a midpoint "fixed-point" ODE solver.
• pendulum_compare.m, [t,u,v]=pendulum_compare(n) compares plots and energies for three pendulum solutions using n steps.
• pendulum_conserved.m, the conserved quantity of the pendulum problem.
• pendulum_deriv.m, the right hand side of the pendulum problem.
• pendulum_euler_backward.m, [t,u,v]=pendulum_euler_backward(n) solves the pendulum ODE using the backward Euler method.
• pendulum_exact.m, the exact solution of the pendulum problem.
• pendulum_midpoint.m, [t,u,v]=pendulum_midpoint(n) solves the pendulum ODE using the midpoint method.
• pendulum_rk4.m, [t,u,v]=pendulum_rk4(n) solves the pendulum ODE using a 4th order Runge-Kutta method.
• predator_conserved.m, h=predator_conserved(rf) evaluates a conserved quantity for the predator-prey ODE system.
• predator_deriv.m, duvdt=predator_deriv(t,rf) defines the derivatives associated with the predator-prey ODE system.
• rk4.m, a fourth order Runge Kutta ODE solver.
• sphere_conserved.m, evaluates the conserved quantity for the sphere problem.
• sphere_deriv.m, the right hand side of the sphere problem.
• sphere_euler.m, solve using Euler's method.
• sphere_midpoint.m, solve using the midpoint method.
• sphere_ode_test.m, runs all the sphere ODE solvers.
• sphere_ode45.m, solve using ode45().
• sphere_rk4.m, solve using Runge Kutta 4.

Images:

• kepler_euler_conserved.png
• kepler_euler_phase.png
• kepler_euler_plot.png
• kepler_midpoint_conserved.png
• kepler_midpoint_phase.png
• kepler_midpoint_plot.png
• kepler_ode45_conserved.png
• kepler_ode45_phase.png
• kepler_ode45_plot.png
• pendulum_compare_plot.png, plots three pendulum solutions using n steps.
• pendulum_compare_energy.png, plots the energy of three pendulum solutions using n steps.
• sphere_euler_plot.png, the Euler solution to the sphere problem.
• sphere_euler_conserved.png, the conserved quantity from the Euler solution.
• sphere_midpoint_plot.png, the midpoint solution to the sphere problem.
• sphere_midpoint_conserved.png, the conserved quantity from the midpoint solution.