numerical methods 4 pde
TRANSCRIPT
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Numerical Methods for Partial
Differential Equations
CAAM 452
Spring 2005
Lecture 41-step time-stepping methods: stability, accuracy
Runge-Kutta Methods,
Instructor: Tim Warburton
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Recall: AB2 v. AB3 v. AB4
These are the margins of absolute stability for the AB methods:
Starting with the yellow AB1 (Euler-Forward) we see that as the order ofaccuracy goes up the stability region shrinks.
i.e. we see that to use the higher order accurate AB scheme we are requiredto take more time steps.
Q) how many more?
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Recall:
Requirements Starting Requirements
AB1:
AB2:
AB3:
1
0
1 1
0
3 1
2 2n n n n
u u dt
u u
u u dt f u f u
!
! !
0
1
0
n n n
u u
u u dt f u
!!
2
1
0
1 1 2
2
0
23 16 512n n n n n
u u dt
u u dt
u u
dtu u f f f
!
!
!
!
1 solution level for start
2 solution levels for start
3 solution levels for start
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cont
So as we take higher order version of the AB
scheme we also need to provide initial values at
more and more levels.
For a problem where we do not know the solution
at more than the initial condition we may have to:
Use AB1 with small dt to get the second restart level
Use AB2 with small dt to get the third restart level
March on using AB3 started with the three levels
achieved above.
AB1
AB2
AB3
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Recall: Derivation of AB Schemes
The AB schemes were motivated by considering
the exactly time integrated ODE:
Which we approximated by using a pth order
polynomial interpolation of the function f
1
1
n
n
t
n n
t
u t u t f u t dt
!
1
1
n
n
t
n n p
t
u t u t I f u t dt
$
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Leap Frog Scheme
We could also have started the integral at:
And used the mid point rule:
Which suggests the leap frog scheme:
1nt
1
1
1 1
n
n
t
n n
t
u t u t f u t dt
!
1 1 2n n nu t u t dtf u t $
1 1 2n n nu u dtf u !
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Volunteer Exercise
1) accuracy: what is the local truncation error?
2) stability: what is the manifold of absolute linear
stability (try analytically) in the nu=dt*mu plane?
2a) what is the region of absolute linear-stability?
1 1 2n n nu u dtf u !
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cont
3) How many starting values are required?
4) Do we have convergence?
5) What is the global order of accuracy?
6) When is this a good method?
1 1 2n n nu u dtf u !
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One Step Methods
Given the difficulties inherent in starting the higher order ABschemes we are encouraged to look for one-step methodswhich only require to evaluate
i.e.
Euler-Forward is a one-step method:
We will consider the one-step Runge-Kutta methods.
For introductory details see: An introduction to numerical analysis, Suli and Mayers, 12.2
(p317) and on
Trefethen p75-
Gustafsson,Kreiss and Oliger p241-
nu 1nu
1 , ;n n n nu u dt u t dt ! *
1 , ; :n n n n n nu u dtf u u t dt f u ! * !
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Runge-Kutta Methods
The Runge-Kutta are a family of one-step methods.
They consist of s stages (i.e. require s evaluations
off)
They will be pth order accurate, for some p.
They are self starting !!!.
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Example Runge-Kutta Method
(Modified Euler)
Modified Euler:
Note how we only need one starting value.
We can also reinterpret this through intermediatevalues:
This looks like a half step to approximate the mid-interval u and then a full step.
This is a 2-stage, 2nd order, single step method.
1
,
,2 2
n n
n n
n n
a dtf u t
a dtb dtf u t
u u b
! !
!
1
1 1 1/ 2
,
2,
n n n
n n n
dtu u f u t
u u dtf u t
!
!
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Linear Stability Analysis
As before we assume that fis linear in u and
independent of time
The scheme becomes (for some given mu):
Which we simplify (eliminate the uhat variable):
1
1 1 1/ 2
,2
,
n n n
n n n
dtu u f u t
u u dtf u t
!
!
1
1 1
2
n n
n n
dtu u u
u u dt u
Q
Q
!
!
1
1 1
2
n n
n n
dtu u u
u u dt u
Q
Q
!
!
2
12
n n n n
dtu u dt u u
QQ !
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cont
We gather all terms on the right hand side:
[ Note: the bracketed term is exactly the first 3
terms of the Taylor series for exp(dt*mu), more on
that later ]
We also note for the numerical solution to be
bounded, and the scheme stable, we require:
2
11
2n n
dtu dt u
QQ
!
2
1 1
2
dtdt
QQ e
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cont
The stability region is the set of nu=mu*dt in the complexplane such that:
The manifold of marginal stability can be found (as in thelinear multistep methods) by fixing the multiplier to be of unitmagnitude and looking for the corresponding values of nuwhich produce this multiplier.
i.e. for each theta find nu such that
2
1 12
RR e
2
1
2
ie
URR !
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cont
We can manually find the roots of this quadratic:
To obtain a parameterized representation of the
manifold of marginal stability:
2
12
ie
URR !
1 1 2 1 ie UR ! s
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Plotting Stability Region for
Modified Euler
1 1 2 1i
e
U
R ! s
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Checking Modified Euler
at the Imaginary Axis
As before we wish to check how much of theimaginary axis is included inside the region of
absolute stability.
Here we plot the real part of the + root
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Is the Imaginary Axis in
the Stability Region ?
We can analytically zoom in by choosingnu=i*alpha (i.e. on the imaginary axis).
We then check the magnitude of the multiplier:
So we know that the only point on the imaginary
axis with multiplier magnitude bounded above by 1is the origin.
Modified Euler is not suitable for the advectionequation.
2 2 22 2 2 4
21 1 1 12 2 2 4
iR E E ER E E ! ! !
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General 2 stage RK family
Consider the four parameter family of RK schemesof the form:
where we will determine the parameters
(a,b,alpha,beta) by consideration of accuracy.
[ Euler-Forward is in this family with a=1,b=0
1
2 1
1 1 2
,
,
n n
n n
n n
k f u t
k f u dtk t dt
u u dt ak bk
F E
!
!
!
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cont
The single step operator in this case is:
1
2 1
1 1 2
1
,
,
,
where , , , ,
n n
n n
n n
n n n n
n n n n n n n n
k f u t
k f u dtk t dt
u u dt ak bk
u u u t
u t af u t bf u dtf u t t dt
F E
F E
!
!
!
p ! *
* !
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cont
We now perform a truncation analysis, similar tothat performed for the linear multistep methods.
We will use the following fact:
2
2
3
3
2 2 2 2
2 2
,
,
...
duf u t t
dt
d u d f f du f f f u t t f
dt dt t u dt t u
d u d f f f
dt dt t u
f f f f f f f f f f f
t t u u t u u t u
!
x x x x ! ! !
x x x x
x x ! ! x x
x x x x x x x x x x x x x x x x
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cont(accuracy)
We expand Phi in terms of powers of dt using thebivariate Taylors expansion
where:
3
2 22 2 2
2 2
, , , ,
2! 2!
n n n n nu t t af u t bf u dtf u t t dt
f
f faf b dt dtf O dt
t u
dt dtf f f f dt dt f t t u u
F E
E F
E FE F
* !
x x ! x x
x x x x x x x
,n n f f u t t !
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cont
We construct the local truncation error as:
Now we choose a,b,alpha,beta to minimize thelocal truncation error.
Note we use subindexing to represent partialderivatives.
22
2 2
2 4
,
2
2 3!
2 2
n n n n n
t u tt tu uu u t u
t u tt tu uu
T u t dt u t dt u t t
dt dt dt f f ff f f f f f f f f f
dt dtf dt af b f dtf dtff f dt ff f O dt
E FE F EF
! *
!
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cont
Consider terms which are the same order in dt in the localtruncation error:
Condition 1:
Condition 2:
Under these conditions, the truncation is order 3 so the
method is 2nd order accurate. It is not possible to further
eliminate the dt^3 terms by adjusting the parameters.
22
2 2
2 4
23
2
2 !
2
n tt tu uu u t u
tt
t u
t u tu uu
dt f ff
d
dtf
f f
T dt f f f f f f f f f
dt dtf dt a b f dt ff tf dtff f O dt E
E F FEF
!
1 0a b !
1 102 2
t u t u f ff b dtf dtff f b bE F E F ! ! !
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Case: No Explicit t Dependence in f
2 23
2
,
2!
n nu t t bf u dtf u
dtff fb f dtf O dt
u u
F
FF
* !
x x!
x x
22 3 2
2
2 3 2,du d u f d u f f f u t f f f
dt dt u dt u ux x x ! ! ! x x x
1
22
2 2 4
;
3! 22
n n n n
uu uuu uu
T u t u t dt u t dt
dtdtdt f f f f dt b f O d dtff f ff tdtf F F
! *
!
11,
2b F ! !
It is easier to generalize to higher order RK in this case when there is no explicit time dependence in f
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Second Example Runge-Kutta:
Heuns Third Order Formula
Traditional version In terms of intermediatevariables:
1
,
,3 3
2 2,
3 3
13
4
n n
n n
n n
n n
a dtf u t
a dtb dtf u t
b dtc dtf u t
u u a c
!
!
!
!
1
2 1 1/3
1 2 2 /3
, 3
2,
3
1
, 3 ,4
n n n
n n
n n n n n
dt
u u f u t
dtu u f u t
u u f u t f u t
!
!
!
This is a 3rd order, 3 stage single step explicit Runge-Kutta method.
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1
2
1
2 3
3
2
3 3
234 3 3
23 1 1
4 3 3
1 2 3
n n
n n n
n n n n n n
n n
n
dtu u u
dt dt u u u u
dt dt dt u u u u u u
dt dt dt u u
dt dt
dt u
Q
Q Q
Q Q Q Q
Q Q Q Q
Q Q
Q
!
!
!
!
!
Again Lets Check the Stability Region
1
2 1 1/3
1 2 2 /3
,3
2,
3
1, 3 ,
4
n n n
n n
n n n n n
dtu u f u t
dtu u f u t
u u f u t f u t
!
!
!
With f=mu*u reduces to a
single level recursion with avery familiar multiplier:
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Stability of Heuns 3rd Order Method
Each marginally stable mu*dt is such that themultiplier is of magnitude 1, i.e.
This traces a curve in the nu=mu*dt complex plane.
Since we are short on time we can plot this using
Matlabs roots function
2 3
1 2 6
i
eUR R
R !
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Stability Region for RK (s=p)
rk2
rk3
rk4
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This time we consider points on the imaginary axiswhich are close to the origin:
And this is bounded above by 1 if
Heuns Method and The Imaginary Axis
22 3
2 22 3
4 6
12 6
12 6
112 36
i
i i
R E
E EE
E EE
E E
!
!
!
3 1.73E e $
rk3
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Observation on RK linear stability
For the sth order, s stage RKwe see that the stability regiongrows with increasing s:
Consequently we can take alarger time step (dt) as theorder of the RK scheme
increase.
On the down side, we requiremore evaluations off
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Popular 4th Order Runge-Kutta Formula
Four stages:
1/ 2
1/ 2
1
1
,
/ 2,
/ 2,
,
1 2 26
n n
n n
n n
n n
n n
a dtf u t
b dtf u a t
c dtf u b t
d dtf u c t
u u a b c d
!
!
!
!
!
see: http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/1all.pdf p76 for details
of minimum number of stages to achieve pth order.
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Imaginary Axis (again)
With the obvious multiplier we obtain:
For stability we require:
22 3 4
22 3 4 6 8
2
12 6 24
1 12 6 24 72 24
i
i i
R E
R R RR
E E E E EE
!
!
!
6 82
28 i.e. 2 2 2.83
72 24
E EE Eu u e $
rk4
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Imaginary Axis Stability Summary
2.83 for the 4th Order Runge-Kutta method
1.73 for Heuns 3rd Order Method
0 for modified Euler
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Bounding the Global Error in Terms of the
Local Truncation Error
Theorem: Consider the general one-step method
and we assume that Phi is Lipschitz continuouswith respect to the first argument (with constant )
i.e. for
Then assuming it follows that
1 , ;n n n nu u dt u t dt ! *
L*
_ a
0 max 0, , , , : ,
we have:
, ; , ;
u t v t D u t t t t u u C
u t dt v t dt L u v*
! e e e
* * e
0 1,2,..,nu t u t C n N e !
00 1
1 , 0,1,..., where maxn L t t
n n nn N
Tu u t e n N T T
L
*
e e *
e ! !
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cont
Proof: we use the definition of the local truncationerror:
to construct the error equation:
we use the Lipschitz continuity of Phi:
tidying:
,n n n n nT u t dt u t dt u t t ! *
1 1 , ,n n n n n n n n nu t u u t u dt u t t u t T ! * *
1 1n n n n n n n
u t u u t u dtL u t u T * e
1 1 1n n n n nu t u dtL u t u T * e
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proofcont
_ a
_ a
_ a
1 1
1 1 1
1
0 0
0
0
1
1 10
1 1
1
1 1
1 1
1
1 1max 1 max
1 1
1 1 1
n n n n n
n n n n
m nn m
n m
m
m nm
n m
m
nm nm
m m
m m nm
n n dtL
u t u dtL u t u T dtL dtL u t u T T
dtL u t u T dtL
T dtL
dtLT dtL T
dtLT T
dtL edtL dtL
*
*
* *
!
* *
!
!
*!
!*
*
e e e e! *
*
* *
e
e
e
e
e !
e e
e
1 0 1n L t t T
e
dtL
*
*
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proof summary
We now have the global error estimate:
Broadly speaking this implies that if the local truncation error
is h^{p+1} then the error at a given time step will scale as
O(h^p):
Convergence follows under restrictions on the ODE which
guarantee existance of a unique C1 solution and stable
choice of dt.
1 01 1 1n L t t n nT
u t u edtL
*
*
e
1 1p
n nu t u O h e
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Warning About Global Error Estimate
It should be noted that the error estimate is ofalmost zero practical use.
In the full convergence analysis we pick a final time
tand we will see that exponential term again.
Convergence is guaranteed but the constant canbe extraordinarily large for finite time:
1 0
1 11n
L t t
n n
Tu t u e
dtL
*
*
e
01
1 L t t
eL
*
*
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A Posteriori Error Estimate
There are examples of RK methods which haveembedded lower order schemes.
i.e. after one full RK time step, for some versions itis possible to use a second set of coefficients to
reconstruct a lower order approximation. Thus we can compute the difference between the
two different approximations to estimate the localtruncation error committed over the time step.
google: runge kutta embedded
Numerical recipes in C:
http://www.library.cornell.edu/nr/bookcpdf/c16-2.pdf
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My Favorite s Stage
Runge-Kutta Method
There is an s stage Runge-Kutta method ofparticular simplicity due to Jameson-Schmidt-
Turkel, which is of interest when there is no explicit
time dependence forf
1
for m=0:s-1
end
n
n
n
u u
dt
u u f us m
u u
!
!
!
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RK v. AB
When should we use RK and when should we useAB?
rk2
rk3
rk4
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Class Cancelled on 02/17/05
There will be no class on Thursday 02/17/05
The homework due for that class will be due the
following Thursday 02/24/05