E19 numerical methods for engineering applications spring

Question

E19: Numerical Methods for Engineering Applications Spring 2016 - HOMEWORK 11

1. Numerical solution of the parachutist problem

The parachutist.py script on the course website presents analytic and numerical (using Euler's method) solutions to the parachutist problem, a classic introduction to differential equations (see, e.g. https://msemac.redwoods.edu/~darnold/math55/deproj/sp04/coleron/paper1.pdf).

In particular this code showcases a reasonable method to solve ODE's analytically using sympy (which might come in handy for your future course work in Engineering). It also shows how to implement Euler's method generically for any ODE of the form

dy/dx = f(x, y)

where x is a scalar, and y is a vector.

a. Extend the program to implement Heun's method (second order RK method with a = b = ½, α = β = 1), as well as the fourth order RK method given by

y_i+1 = y_i + h/6(k₁ + 2k₂ + 2k₃ + k₁)

where

k₁ = f(x_i, y_i)

k₂ = f(x_i + h/2, y_i + (h/2)k₁)

k₃ = f(x_i + h/2, y_i + (h/2)k₂)

k₄ = f(x_i + h, y_i + hk₃)

b. Plot the numerical solutions with a step size of h = 1 against the analytic solution.

Attachment:- Assignment.rar