Solving partial differential equations (PDEs)
Hans Fangohr
Engineering and the EnvironmentUniversity of Southampton
United [email protected]
May 3, 2012
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Outline I
1 Introduction: what are PDEs?
2 Computing derivatives using finite differences
3 Diffusion equation
4 Recipe to solve 1d diffusion equation
5 Boundary conditions, numerics, performance
6 Finite elements
7 Summary
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This lecture
tries to compress several years of material into 45 minutes
has lecture notes and code available for download athttp://www.soton.ac.uk/fangohr/teaching/comp6024
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http://www.soton.ac.uk/~fangohr/teaching/comp6024
What are partial differential equations (PDEs)
Ordinary Differential Equations (ODEs)one independent variable, for example t in
d2x
dt2= k
mx
often the indepent variable t is the timesolution is function x(t)important for dynamical systems, population growth,control, moving particles
Partial Differential Equations (ODEs)multiple independent variables, for example t, x and y in
u
t= D
(2u
x2+
2u
y2
)solution is function u(t, x, y)important for fluid dynamics, chemistry,electromagnetism, . . . , generally problems with spatialresolution
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2d Diffusion equation
u
t= D
(2u
x2+2u
y2
)u(t, x, y) is the concentration [mol/m3]
t is the time [s]
x is the x-coordinate [m]
y is the y-coordinate [m]
D is the diffusion coefficient [m2/s]
Also known as Ficks second law. The heat equation has thesame structure (and u represents the temperature).Example:http://www.youtube.com/watch?v=WC6Kj5ySWkQ
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http://www.youtube.com/watch?v=WC6Kj5ySWkQ
Examples of PDEs
Cahn Hilliard Equation (phase separation)
Fluid dynamics (including ocean and atmospheric models,plasma physics, gas turbine and aircraft modelling)Structural mechanics and vibrations, superconductivity,micromagnetics, . . .
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http://en.wikipedia.org/wiki/CahnHilliard_equation
Computing derivatives
using finite differences
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Overview
Motivation:
We need derivatives of functions for example foroptimisation and root finding algorithmsNot always is the function analytically known (but weare usually able to compute the function numerically)The material presented here forms the basis of thefinite-difference technique that is commonly used tosolve ordinary and partial differential equations.
The following slides show
the forward difference techniquethe backward difference technique and thecentral difference technique to approximate thederivative of a function.We also derive the accuracy of each of these methods.
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The 1st derivative
(Possible) Definition of the derivative (or differentialoperator d
dx)
f (x) dfdx
(x) = limh0
f(x+ h) f(x)h
Use difference operator to approximate differentialoperator
f (x) =df
dx(x) = lim
h0
f(x+ h) f(x)h
f(x+ h) f(x)h
can now compute an approximation of f (x) simply byevaluating f (twice).
This is called the forward difference because we use f(x)and f(x+ h).
Important questions: How accurate is this approximation?
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Accuracy of the forward difference
Formal derivation using the Taylor series of f around x
f(x + h) =
n=0
hnf (n)(x)
n!
= f(x) + hf (x) + h2f (x)
2!+ h3
f (x)
3!+ . . .
Rearranging for f (x)
hf (x) = f(x + h) f(x) h2 f(x)
2! h3 f
(x)
3! . . .
f (x) =1
h
(f(x + h) f(x) h2 f
(x)
2! h3 f
(x)
3! . . .
)=
f(x + h) f(x)h
h2 f
(x)2! h
3 f(x)3!
h . . .
=f(x + h) f(x)
h hf
(x)
2! h2 f
(x)
3! . . .
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Accuracy of the forward difference (2)
f (x) =f(x+ h) f(x)
h hf
(x)
2! h2f
(x)
3! . . .
Eforw(h)
f (x) =f(x+ h) f(x)
h+ Eforw(h)
Therefore, the error term Eforw(h) is
Eforw(h) = hf (x)
2! h2f
(x)
3! . . .
Can also be expressed as
f (x) =f(x+ h) f(x)
h+O(h)
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The 1st derivative using the backward difference
Another definition of the derivative (or differentialoperator d
dx)
df
dx(x) = lim
h0
f(x) f(x h)h
Use difference operator to approximate differentialoperator
df
dx(x) = lim
h0
f(x) f(x h)h
f(x) f(x h)h
This is called the backward difference because we usef(x) and f(x h).How accurate is the backward difference?
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Accuracy of the backward difference
Formal derivation using the Taylor Series of f around x
f(x h) = f(x) hf (x) + h2f(x)
2! h3f
(x)
3!+ . . .
Rearranging for f (x)
hf (x) = f(x) f(x h) + h2 f(x)
2! h3 f
(x)
3! . . .
f (x) =1
h
(f(x) f(x h) + h2 f
(x)
2! h3 f
(x)
3! . . .
)=
f(x) f(x h)h
+h2 f
(x)2! h
3 f(x)3!
h . . .
=f(x) f(x h)
h+ h
f (x)
2! h2 f
(x)
3! . . .
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Accuracy of the backward difference (2)
f (x) =f(x) f(x h)
h+ h
f (x)
2! h2f
(x)
3! . . .
Eback(h)
f (x) =f(x) f(x h)
h+ Eback(h) (1)
Therefore, the error term Eback(h) is
Eback(h) = hf (x)
2! h2f
(x)
3! . . .
Can also be expressed as
f (x) =f(x) f(x h)
h+O(h)
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Combining backward and forward differences (1)
The approximations are
forward:
f (x) =f(x+ h) f(x)
h+ Eforw(h) (2)
backward
f (x) =f(x) f(x h)
h+ Eback(h) (3)
Eforw(h) = hf (x)
2! h2 f
(x)
3! h3 f
(x)
4! h4 f
(x)
5! . . .
Eback(h) = hf (x)
2! h2 f
(x)
3!+ h3
f (x)
4! h4 f
(x)
5!+ . . .
Add equations (2) and (3) together, then the error cancelspartly!
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Combining backward and forward differences (2)
Add these lines together
f (x) =f(x + h) f(x)
h+ Eforw(h)
f (x) =f(x) f(x h)
h+ Eback(h)
2f (x) =f(x + h) f(x h)
h+ Eforw(h) + Eback(h)
Adding the error terms:
Eforw(h) + Eback(h) = 2h2f (x)
3! 2h4 f
(x)
5! . . .
The combined (central) difference operator is
f (x) =f(x + h) f(x h)
2h+ Ecent(h)
with
Ecent(h) = h2f (x)
3! h4 f
(x)
5! . . .
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Central difference
Can be derived (as on previous slides) by adding forwardand backward difference
Can also be interpreted geometrically by defining thedifferential operator as
df
dx(x) = lim
h0
f(x+ h) f(x h)2h
and taking the finite difference form
df
dx(x) f(x+ h) f(x h)
2h
Error of the central difference is only O(h2), i.e. betterthan forward or backward difference
It is generally the case that symmetric differencesare more accurate than asymmetric expressions.
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Example (1)
Using forward difference to estimate the derivative off(x) = exp(x)
f (x) f forw =f(x+ h) f(x)
h=
exp(x+ h) exp(x)h
Numerical example:
h = 0.1, x = 1
f (1) f forw(1.0) =exp(1.1)exp(1)
0.1= 2.8588
Exact answers is f (1.0) = exp(1) = 2.71828
(Central diff: f cent(1.0) =exp(1+0.1)exp(10.1)
0.2 = 2.72281)
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Example (2)
Comparison: forward difference, central difference and exactderivative of f(x) = exp(x)
0 1 2 3 4 5x
0
20
40
60
80
100
120
140
df/
dx(x
)
Approximations of df/dx for f(x)=exp(x)
forward h=1central h=1forward h=0.0001exact
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Summary
Can approximate derivatives of f numerically using onlyfunction evaluations of f
size of step h very important
central differences has smallest error term
name formula error
forward f (x) = f(x+h)f(x)h
O(h)backward f (x) = f(x)f(xh)
hO(h)
central f (x) = f(x+h)f(xh)2h
O(h2)
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Appendix: source to compute figure on page 19 I
EPS=1 #very large EPS to provoke inaccuracy
def forwarddiff(f,x,h=EPS):
# df/dx = ( f(x+h)-f(x) )/h + O(h)
return ( f(x+h)-f(x) )/h
def backwarddiff(f,x,h=EPS):
# df/dx = ( f(x)-f(x-h) )/h + O(h)
return ( f(x)-f(x-h) )/h
def centraldiff(f,x,h=EPS):
# df/dx = (f(x+h) - f(x-h))/h + O(h^2)
return (f(x+h) - f(x-h))/(2*h)
if __name__ == "__main__":
#create example plot
import pylab
import numpy as np
a=0 #left and
b=5 #right limits for x
N=11 #steps
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Appendix: source to compute figure on page 19 II
def f(x):
""" Our test funtion with
convenient property that
df/dx = f"""
return np.exp(x)
xs=np.linspace(a,b,N)
forward = []
forward_small_h = []
central = []
for x in xs:
forward.append( forwarddiff(f,x) )
central.append( centraldiff(f,x) )
forward_small_h.append(
forwarddiff(f,x,h=1e-4))
pylab.figure(figsize =(6 ,4))
pylab.axis([a,b,0,np.exp(b)])
pylab.plot(xs,forward ,^,label=forward h=%g%EPS)
pylab.plot(xs,central ,x,label=central h=%g%EPS)
p
