# 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

## 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.

## 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 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.

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