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

Goal

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Problem setting and methods

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

$$ \begin{align} u'(t) &= -au(t) \label{ode}\\ u(0) &= I \label{initial:value} \end{align} $$

Here,

$ t\in (0,T] $
$ a $, $ I $, and $ T $ are prescribed parameters
$ u(t) $ is the unknown function
The ODE \eqref{ode} has the initial condition \eqref{initial:value}

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.