1. Numerical Methods for Partial Differential Equations
CAAM 452
Spring 2005
Lecture 4
1-step time-stepping methods: stability, accuracy
Runge-Kutta Methods,
Instructor: Tim Warburton

2. Today Recall AB stability regions and start up issues
Group analysis of the Leap-Frog scheme
One-step methods
Example Runge-Kutta methods:
Modified Euler
General family of 2nd order RK methods
Heun’s 3rd Order Method
The 4th Order “Runge-Kutta” method
Jameson-Schmidt-Turkel
Linear Absolute Stability Regions for the 2nd order family RK
Global error analysis for general 1-step methods (stops slightly short of a full convergence analysis)
Warning on usefulness of global error estimate
Discussion on AB v. RK
Embedded lower order RK schemes useful for a posteriori error estimates.
CAAM 452 Spring 2005

3. 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 of accuracy goes up the stability region shrinks.
i.e. we see that to use the higher order accurate AB scheme we are required to take more time steps.
Q) how many more?
CAAM 452 Spring 2005

4. Recall: Requirements Starting Requirements
AB1:
AB2:
AB3:
1 solution level for start
2 solution levels for start
3 solution levels for start
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5. 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.
AB2
AB1
AB3
CAAM 452 Spring 2005

6. Recall: Derivation of AB Schemes
The AB schemes were motivated by considering the exactly time integrated ODE:
Which we approximated by using a p’th order polynomial interpolation of the function f
CAAM 452 Spring 2005

7. Leap Frog Scheme We could also have started the integral at:
And used the mid point rule:
Which suggests the leap frog scheme:
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8. Volunteer Exercise 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?
CAAM 452 Spring 2005

9. 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?
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10. One Step Methods
Given the difficulties inherent in starting the higher order AB schemes we are encouraged to look for one-step methods which 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-
CAAM 452 Spring 2005

11. Runge-Kutta Methods The Runge-Kutta are a family of one-step methods.
They consist of s stages (i.e. require s evaluations of f)
They will be p’th order accurate, for some p.
They are self starting !!!.
CAAM 452 Spring 2005

12. Example Runge-Kutta Method (Modified Euler)
Note how we only need one starting value.
We can also reinterpret this through “intermediate” values:
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.
CAAM 452 Spring 2005

13. Linear Stability Analysis
As before we assume that f is linear in u and independent of time
The scheme becomes (for some given mu):
Which we simplify (eliminate the uhat variable):
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14. 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:
CAAM 452 Spring 2005

15. cont
The stability region is the set of nu=mu*dt in the complex plane such that:
The manifold of marginal stability can be found (as in the linear multistep methods) by fixing the multiplier to be of unit magnitude and looking for the corresponding values of nu which produce this multiplier.
i.e. for each theta find nu such that
CAAM 452 Spring 2005

16. cont We can manually find the roots of this quadratic:
To obtain a parameterized representation of the manifold of marginal stability:
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17. Plotting Stability Region for Modified Euler
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18. Checking Modified Euler at the Imaginary Axis
As before we wish to check how much of the imaginary axis is included inside the region of absolute stability.
Here we plot the real part of the “+” root
CAAM 452 Spring 2005

19. Is the Imaginary Axis in the Stability Region ?
We can analytically zoom in by choosing nu=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 1 is the origin.
Modified Euler is not suitable for the advection equation.
CAAM 452 Spring 2005

20. General 2 stage RK family
Consider the four parameter family of RK schemes of 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
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21. cont
The single step operator in this case is:
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22. cont
We now perform a truncation analysis, similar to that performed for the linear multistep methods.
We will use the following fact:
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23. cont (accuracy)
We expand Phi in terms of powers of dt using the bivariate Taylor’s expansion
where:
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24. cont We construct the local truncation error as:
Now we choose a,b,alpha,beta to minimize the local truncation error.
Note – we use subindexing to represent partial derivatives.
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25. cont
Consider terms which are the same order in dt in the local truncation 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.
CAAM 452 Spring 2005

26. Case: No Explicit t Dependence in f
It is easier to generalize to higher order RK in this case when there is no explicit time dependence in f.
CAAM 452 Spring 2005

27. Second Example Runge-Kutta: Heun’s Third Order Formula
Traditional version
In terms of intermediate variables:
This is a 3rd order, 3 stage single step explicit Runge-Kutta method.
CAAM 452 Spring 2005

28. Again Let’s Check the Stability Region
With f=mu*u reduces to a
single level recursion with a
very familiar multiplier:
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29. Stability of Heun’s 3rd Order Method
Each marginally stable mu*dt is such that the multiplier 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 Matlab’s roots function…
CAAM 452 Spring 2005

30. Stability Region for RK (s=p)
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31. Heun’s Method and The Imaginary Axis
This time we consider points on the imaginary axis which are close to the origin:
And this is bounded above by 1 if
rk3
CAAM 452 Spring 2005

32. Observation on RK linear stability
For the s’th order, s stage RK we see that the stability region grows with increasing s:
Consequently we can take a larger time step (dt) as the order of the RK scheme increase.
On the down side, we require more evaluations of f
CAAM 452 Spring 2005

33. Popular 4th Order Runge-Kutta Formula
Four stages:
see: p76 for details
of minimum number of stages to achieve p’th order.
CAAM 452 Spring 2005

34. Imaginary Axis (again)
With the obvious multiplier we obtain:
For stability we require:
rk4
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35. Imaginary Axis Stability Summary
2.83 for the 4th Order “Runge-Kutta” method
1.73 for Heun’s 3rd Order Method
0 for modified Euler
CAAM 452 Spring 2005

36. 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 continuous with respect to the first argument (with constant )
i.e. for
Then assuming it follows that
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37. cont Proof: we use the definition of the local truncation error:
to construct the error equation:
we use the Lipschitz continuity of Phi:
tidying:
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38. proof cont
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39. 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.
CAAM 452 Spring 2005

40. Warning About Global Error Estimate
It should be noted that the error estimate is of almost zero practical use.
In the full convergence analysis we pick a final time t and we will see that exponential term again.
Convergence is guaranteed but the constant can be extraordinarily large for finite time:
CAAM 452 Spring 2005

41. A Posteriori Error Estimate
There are examples of RK methods which have embedded lower order schemes.
i.e. after one full RK time step, for some versions it is 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 local truncation error committed over the time step.
google: runge kutta embedded
Numerical recipes in C:
CAAM 452 Spring 2005

42. My Favorite s Stage Runge-Kutta Method
There is an s stage Runge-Kutta method of particular simplicity due to Jameson-Schmidt-Turkel, which is of interest when there is no explicit time dependence for f
CAAM 452 Spring 2005

43. RK v. AB When should we use RK and when should we use AB? rk3 rk2 rk4
CAAM 452 Spring 2005

44. 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
CAAM 452 Spring 2005