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Harvard Applied Math 205: Code Examples
Unit 3: Numerical Calculus
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quadrat.py - Integrates a function using composite trapezoid and Simpson's
rules. Outputs the error as a function of the interval length h.

diff.py - Constructs a differentiation matrix, and uses it to calculate the
derivative of a test function.

l-v.py - Demonstrates Python's black box integration routine odeint in the
scipy.integrate module. Solves the two-component Lotka–Volterra model for
predator–prey interactions.

euler.py - Demonstrates the simple first-order forward Euler method for
integrating an ordinary differential equation.

e_stab.py - Tests the stability of the choice of timestep in the forward Euler
method

order2.py - Demonstrates three different second-order time-integration
schemes: the modified Euler method, the improved Euler method, and Ralston's
method.

r_extrap.py - Demonstrates applying Richardson extrapolation to the forward
Euler method, to achieve a new method of second order.

r_extrap2.py - Demonstrates applying Richardson extrapolation twice to the
forward Euler method, to achieve a new method of third order.

stiff.py - Solves a two-component differential equation that has one stiff
mode. It uses an explict scheme and an implicit scheme. The explicit scheme
has a very strict timestep restriction, while the implicit scheme is
unconditionally stable.

stiff2.py - Demonstrates using Python's black-box stiff solver for the
same two-component differential equation from stiff.py.

ad-bash.py - Solves a differential equation using a second-order
Adams-Bashforth multi-step scheme.

ode_bvp.py - Solves an ODE boundary value problem by formulating it as a
linear system using finite differences.

transp.py - Solves the linear advection equation using a first-order scheme
with an upwinded one-sided derivative.

transp2.py - Solves the linear advection equation where the spatial derivative
is calculated using a second-order centered-difference scheme. This scheme is
slightly unstable, which becomes apparant after many timesteps have been
performed.

m_of_lines.py - Demonstrates the method of lines for the linear advection
equation, whereby the spatial grid is discretized but the system is
time-integrated using a black-box ODE integration routine.

heat.py - Solves the heat equation on a periodic unit interval using an
explicit finite difference scheme.

c-n.py - Solves the heat equation using the Crank-Nicolson method, which is
second-order in space and time, and unconditionally stable.

poisson.py - Solves the two-dimensional Poisson equation using a linear solve
of a finite-difference discretization.