Solving ODEs Euler Method & RK2/4

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1 Solving ODEs Euler Method & RK2/4 Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM 02/11/10 1

2 Simulation example: Formation of Stars SPH simulation with gravity and supersonic turbulence. Initial conditions: Uniform density 1000M sun 1pc diameter Temperature 10K

3 Magneto Rotational Instability (MRI) drives turbulence in accretion disks 3 Simulation by Mario Flock using the Pluto code. 12/13/2009 Hubert Klahr - Planet Formation - MPIA Heidelberg

Formation Of Planetesimals From pressure trapped / gravitational Bound heaps of gravel - here magnetic turbulence: Gravoturbulent formation of planetesimals - Concentration in Zonal Flows:

4 Formation Of Planetesimals From pressure trapped / gravitational Bound heaps of gravel - here magnetic turbulence: Gravoturbulent formation of planetesimals - Concentration in Zonal Flows: Johansen, Klahr & Henning Vortices: Raettig, Klahr & Lyra 512 ^2 simulation 64 Mio particles Entire project used 15 Mio. CPU hours. 12/13/2009 Hubert Klahr - Planet Formation - MPIA Heidelberg 4

5 Local collaps 100 x resolution: 0.1 Roche; St: 0.1 Dittrich 2013 PhD thesis 5

6 MHD plus self-gravity for the dust, including particle feed back on the gas: Pencil Code: Finite Differenzen / Runge-Kutta 5th order! gas dust Poisson equation solved via FFT in parallel mode: up to cells 12/13/2009 Hubert Klahr - Planet Formation - MPIA Heidelberg

7 Euler Method

8 Euler s Method y True value y 1, Predicted Slope x 0,y 0 Φ value Step size, h x Figure 1 Graphical interpretation of the first step of Euler s method 3

9 Euler s Method y True Value y i+1, Predicted value Φ y i h Step size x i x i+1 x 4 Figure 2. General graphical interpretation of Euler s method

10 How to write Ordinary Differential Equation How does one write a first order differential equation in the form of Example is rewritten as In this case 5

11 Example A 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. 6

12 Solution Step 1: is the approximate temperature at 7

13 Solution Cont Step 2: For is the approximate temperature at 8

14 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 9

15 Comparison of Exact and Numerical Solutions 1200,0000 Temperature, θ(k) 900, , ,0000 Exact Solution h=240 0, Time, t(sec) Figure 3. Comparing exact and Euler s method 10

16 Effect of step size Table 1. Temperature at 480 seconds as a function of step size, h Step, h θ(480) E t є t % (exact)

17 Comparison with exact results 1200, ,0000 Exact solution θ(k) Temperature, 600, ,0000 0, , ,0000 h=120 h=480 h= , , Time, t (sec) Figure 4. Comparison of Euler s method with exact solution for different step sizes 12

18 Effects of step size on Euler s Method 750, ,0000 Temperature, θ(k) 250,0000 0, , , , , Step size, h (s) 13 Figure 5. Effect of step size in Euler s method.

19 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 are the Euler s method. The true error in the approximation is given by 14

20 Runge 2 nd Order Method Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates 02/11/10 1

21 Runge-Kutta 2 nd Order Method

22 Runge-Kutta 2 nd Order Method For Runge Kutta 2nd order method is given by where 3

23 Heun s Method Heun s method Here a 2 =1/2 is chosen y y i+1, predicted resulting in y i where x i x i+1 Figure 1 Runge-Kutta 2nd order method (Heun s method) x 4

24 Midpoint Method Here is chosen, giving resulting in where 5

25 Ralston s Method Here is chosen, giving resulting in where 6

26 How to write Ordinary Differential Equation How does one write a first order differential equation in the form of Example is rewritten as In this case 7

27 Example A 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 Heun s method. Assume a step size of seconds. 8

28 Solution Step 1: 9

29 Solution Cont Step 2: 10

30 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 11

31 Comparison with exact results Figure 2. Heun s method results for different step sizes 12

32 Effect of step size Table 1. Temperature at 480 seconds as a function of step size, h Step size, h θ(480) E t є t % (exact) 13

33 Effects of step size on Heun s Method Figure 3. Effect of step size in Heun s method 14

34 Step size, h Comparison of Euler and Runge- Kutta 2 nd Order Methods Table 2. Comparison of Euler and the Runge-Kutta methods θ(480) Euler Heun Midpoint Ralston (exact)

35 Comparison of Euler and Runge- Kutta 2 nd Order Methods Table 2. Comparison of Euler and the Runge-Kutta methods Step size, h Euler Heun Midpoint Ralston (exact)

36 Comparison of Euler and Runge- Kutta 2 nd Order Methods 1200,0000 Temperature, θ(k) 1025, ,0000 Analytical Midpoint Ralston Heun 675,0000 Euler , Time, t (sec) Figure 4. Comparison of Euler and Runge Kutta 2 nd order methods with exact results.

37 Additional Resources For 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 /topics/ runge_kutta_2nd_method.html

38 Runge 4 th Order Method Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates 02/11/10 1

39 Runge-Kutta 4 th Order Method

40 Runge-Kutta 4 th Order Method For Runge Kutta 4 th order method is given by where 3

41 How to write Ordinary Differential Equation How does one write a first order differential equation in the form of Example is rewritten as In this case 4

42 Example A 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 Assume a step size of seconds using Runge-Kutta 4 th order method. seconds. 5

43 Solution Step 1: 6

44 Solution Cont is the approximate temperature at 7

45 Solution Cont Step 2: 8

46 Solution Cont θ 2 is the approximate temperature at 9

47 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 10

48 Comparison with exact results 11 Figure 1. Comparison of Runge-Kutta 4th order method with exact solution

49 Effect of step size Table 1. Temperature at 480 seconds as a function of step size, h Step size, h θ (480) E t є t % (exact)

50 Effects of step size on Runge- Kutta 4 th Order Method 13 Figure 2. Effect of step size in Runge-Kutta 4th order method

51 Comparison of Euler and Runge- Kutta Methods 14 Figure 3. Comparison of Runge-Kutta methods of 1st, 2nd, and 4th order.

52 MHD plus self-gravity for the dust, including particle feed back on the gas: Pencil Code: Finite Differenzen / Runge-Kutta 5th order! gas dust Poisson equation solved via FFT in parallel mode: up to cells 12/13/2009 Hubert Klahr - Planet Formation - MPIA Heidelberg

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Romberg Rule of Integration

Romberg Rule of Integration Romberg Rule of ntegration Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates 1/10/2010 1 Romberg Rule of ntegration Basis