Solving ODEs Euler Method & RK2/4klahr/Lecture_03_OdE.pdfDittrich 2013 PhD thesis 5 Local collaps 100 x resolution: 0.1 Roche; St: 0.1 12/13/2009 Hubert Klahr - Planet Formation -

02/11/10 http://numericalmethods.eng.usf.edu 1

Solving ODEs Euler Method & RK2/4

Major: All Engineering Majors

Authors: Autar Kaw, Charlie Barker

http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM

http://www.mpia.de/homes/mordasini/UKNUM/UeB_ode.pdf

http://numericalmethods.eng.usf.edu/

http://www.mpia.de/homes/mordasini/UKNUM/UeB_ode.pdf

Simulation example: Formation of StarsSPH simulation with gravity and super- sonic turbulence.

Initial conditions:

Uniform density 1000Msun

1pc diameter Temperature 10K

12/13/2009 Hubert Klahr - Planet Formation - MPIA Heidelberg

3

Magneto Rotational Instability (MRI) drives turbulence in accretion disks

Simulation by Mario Flock using the Pluto code.


4

Formation Of Planetesimals From pressure trapped /gravitational Bound heaps of gravel - here magnetic turbulence:

Johansen, Klahr & Henning 2011. Vortices: Raettig, Klahr & Lyra

512 ^2 simulation 64 Mio particles Entire project used 15 Mio. CPU hours.

Gravoturbulent formation of planetesimals - Concentration in Zonal Flows:

5Dittrich 2013 PhD thesis

Local collaps 100 x resolution: 0.1 Roche; St: 0.1


gas

dust

MHD plus self-gravity for the dust, including particle feed back on the gas: Pencil Code: Finite Differenzen / Runge-Kutta 5th order!

Poisson equation solved via FFT in parallel mode: up to 2563 cells

Euler Method

http://numericalmethods.eng.usf.edu


http://numericalmethods.eng.usf.edu3

Euler’s Method

Φ

Step size, h

x

y

x0,y0

True value

y1, Predicted valueSlope

Figure 1 Graphical interpretation of the first step of Euler’s method


Euler’s Method

Φ

Step size

h

True Value

yi+1, Predicted value

yi

x

y

xi xi+1

Figure 2. General graphical interpretation of Euler’s method


How to write Ordinary Differential Equation

Example

is rewritten as

In this case

How does one write a first order differential equation in the form of


ExampleA ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by

Find the temperature at seconds using Euler’s method. Assume a step size of

seconds.


SolutionStep 1:

is the approximate temperature at


Solution ContFor Step 2:



Solution Cont

The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as

The solution to this nonlinear equation at t=480 seconds is


Comparison of Exact and Numerical Solutions

Figure 3. Comparing exact and Euler’s method

Tem

pera

ture

,

0,0000

300,0000

600,0000

900,0000

1200,0000

Time, t(sec)0 125 250 375 500

h=240

Exact Solution

θ(K

)

Step, h θ(480) Et |єt|%

480 240 120 60 30

−987.81 110.32 546.77 614.97

1635.4 537.26 100.80 32.607 14.806

252.54 82.964 15.566 5.0352 2.2864


Effect of step size

Table 1. Temperature at 480 seconds as a function of step size, h

(exact)


Comparison with exact resultsTe

mpe

ratu

re,

-1200,0000

-900,0000

-600,0000

-300,0000

0,0000

300,0000

600,0000

900,0000

1200,0000

Time, t (sec)0 125 250 375 500

Exact solution

h=120h=240

h=480

θ(K

)

Figure 4. Comparison of Euler’s method with exact solution for different step sizes


Effects of step size on Euler’s Method

Tem

pera

ture

,

-1000,0000

-750,0000

-500,0000

-250,0000

0,0000

250,0000

500,0000

750,0000

Step size, h (s) 0 125 250 375 500

θ(K

)

Figure 5. Effect of step size in Euler’s method.


Errors in Euler’s Method

It can be seen that Euler’s method has large errors. This can be illustrated using Taylor series.

As you can see the first two terms of the Taylor series

The true error in the approximation is given by

are the Euler’s method.


Runge 2nd Order Method




Undergraduates


Runge-Kutta 2nd Order Method




Runge-Kutta 2nd Order Method

Runge Kutta 2nd order method is given by

where

For


Heun’s Method

x

y

xi xi+1

yi+1, predicted

yi

Figure 1 Runge-Kutta 2nd order method (Heun’s method)

Heun’s method

resulting in

where

Here a2=1/2 is chosen


Midpoint MethodHere is chosen, giving

resulting in

where


Ralston’s MethodHere is chosen, giving

resulting in

where



Example

is rewritten as

In this case




Find the temperature at seconds using Heun’s method. Assume a step size of

seconds.


SolutionStep 1:


Solution ContStep 2:


Solution Cont




Comparison with exact results

Figure 2. Heun’s method results for different step sizes


Effect of step sizeTable 1. Temperature at 480 seconds as a function of step size, h

Step size, h θ(480) Et |єt|%

480 240 120 60 30

−393.87 584.27 651.35 649.91 648.21

1041.4 63.304 −3.7762 −2.3406 −0.63219

160.82 9.7756 0.58313 0.36145 0.097625

(exact)


Effects of step size on Heun’s Method

Figure 3. Effect of step size in Heun’s method

Step size, h

θ(480)

Euler Heun Midpoint Ralston480 240 120 60

−987.84 110.32 546.77 614.97

−393.87 584.27 651.35 649.91

1208.4 976.87 690.20 654.85

449.78 690.01 667.71 652.25


Comparison of Euler and Runge-Kutta 2nd Order Methods

Table 2. Comparison of Euler and the Runge-Kutta methods

(exact)



Table 2. Comparison of Euler and the Runge-Kutta methods

Step size, h Euler Heun Midpoint Ralston

480 240 120 60 30

252.54 82.964 15.566 5.0352 2.2864

160.82 9.7756 0.58313 0.36145 0.097625

86.612 50.851 6.5823 1.1239 0.22353

30.544 6.5537 3.1092 0.72299 0.15940

(exact)



Figure 4. Comparison of Euler and Runge Kutta 2nd order methods with exact results.

Tem

pera

ture

,

500,0000

675,0000

850,0000

1025,0000

1200,0000

Time, t (sec)0 125 250 375 500

Analytical

Ralston

Midpoint

Euler

Heun

θ(K

)

Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/runge_kutta_2nd_method.html




Runge 4th Order Method




Undergraduates


Runge-Kutta 4th Order Method




Runge-Kutta 4th Order Method

where

For

Runge Kutta 4th order method is given by



Example

is rewritten as

In this case




Find the temperature at seconds using Runge-Kutta 4th order method.

seconds.Assume a step size of


SolutionStep 1:


Solution Cont



Solution ContStep 2:


Solution Cont

θ2 is the approximate temperature at


Solution Cont




Comparison with exact results

Figure 1. Comparison of Runge-Kutta 4th order method with exact solution

Step size, h

θ (480) Et |єt|%

480 240 120 60 30

−90.278 594.91 646.16 647.54 647.57

737.85 52.660 1.4122 0.033626 0.00086900

113.94 8.1319 0.21807 0.0051926 0.00013419


Effect of step sizeTable 1. Temperature at 480 seconds as a function of step size, h

(exact)


Effects of step size on Runge-Kutta 4th Order Method

Figure 2. Effect of step size in Runge-Kutta 4th order method


Comparison of Euler and Runge-Kutta Methods

Figure 3. Comparison of Runge-Kutta methods of 1st, 2nd, and 4th order.


gas

dust

MHD plus self-gravity for the dust, including particle feed back on the gas: Pencil Code: Finite Differenzen / Runge-Kutta 5th order!

Poisson equation solved via FFT in parallel mode: up to 2563 cells

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Documents

Solving ODEs Euler Method & RK2/4klahr/Lecture_03_OdE.pdfDittrich 2013 PhD thesis 5 Local collaps 100 x resolution: 0.1 Roche; St: 0.1 12/13/2009 Hubert Klahr - Planet Formation -