On Schemes for Exponential Decay

Hans Petter Langtangen [1, 2] (hpl at simula.no)

 

[1] Center for Biomedical Computing, Simula Research Laboratory
[2] Department of Informatics, University of Oslo

 

Sep 24, 2015


© 2015, Hans Petter Langtangen. Released under CC Attribution 4.0 license

Goal

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Problem setting and methods

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

 
$$ \begin{align} u'(t) &= -au(t) \tag{1}\\ u(0) &= I \tag{2} \end{align} $$

 

Here,

  • \( t\in (0,T] \)
  • \( a \), \( I \), and \( T \) are prescribed parameters
  • \( u(t) \) is the unknown function
  • The ODE (1) has the initial condition (2)

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

Implementation in a Python function:

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

A complete main program.

# 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 (\( \theta=1/2 \))

Concluding remarks:

Only the Backward Euler scheme is guaranteed to always give qualitatively correct results.