Goal

The primary goal of this demo talk is to demonstrate how to write talks with DocOnce and get them rendered in numerous HTML formats.

Problem setting and methods

We aim to solve the (almost) simplest possible differential equation problem

The ODE problem is solved by a finite difference scheme

• Mesh in time: $0= t_0 < t_1 \cdots < t_N=T$
• Assume constant $\Delta t = t_{n}-t_{n-1}$
• $u^n$: numerical approx to the exact solution at $t_n$

The $\theta$ rule,  
 
$$u^{n+1} = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}u^n, \quad n=0,1,\ldots,N-1$$  
 
contains the Forward Euler ($\theta=0$), the Backward Euler ($\theta=1$), and the Crank-Nicolson ($\theta=0.5$) schemes.

The Forward Euler scheme explained

Implementation

def solver(I, a, T, dt, theta):
    """Solve u'=-a*u, u(0)=I, for t in (0,T]; step: dt."""
    dt = float(dt)           # avoid integer division
    N = int(round(T/dt))     # no of time intervals
    T = N*dt                 # adjust T to fit time step dt
    u = zeros(N+1)           # array of u[n] values
    t = linspace(0, T, N+1)  # time mesh

    u[0] = I                 # assign initial condition
    for n in range(0, N):    # n=0,1,...,N-1
        u[n+1] = (1 - (1-theta)*a*dt)/(1 + theta*dt*a)*u[n]
    return u, t

How to use the solver function

# Set problem parameters
I = 1.2
a = 0.2
T = 8
dt = 0.25
theta = 0.5

from solver import solver, exact_solution
u, t = solver(I, a, T, dt, theta)

import matplotlib.pyplot as plt
plt.plot(t, u, t, exact_solution)
plt.legend(['numerical', 'exact'])
plt.show()

Results

The Crank-Nicolson method shows oscillatory behavior for not sufficiently small time steps, while the solution should be monotone

The artifacts can be explained by some theory

Exact solution of the scheme:  
 
$$u^n = A^n,\quad A = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}\thinspace .$$  
 

Key results:

• Stability: $|A| < 1$
• No oscillations: $A>0$
• $\Delta t < 1/a$ for Forward Euler ($\theta=0$)
• $\Delta t < 2/a$ for Crank-Nicolson ($\theta=1/2$)