Hampiran Nuemik Solusi Persamaan Differensial (Lanjutan) Pertemuan 11

1. Hampiran Nuemik Solusi Persamaan Differensial (Lanjutan)Pertemuan 11 Matakuliah : K0342 / Metode Numerik I Tahun : 2006

2. PERTEMUAN-11 Hampiran Nuemik Solusi Persamaan Differensial (Lanjutan) Runge Kutta

18. Runge-Kutta Methods Runge-Kutta methods are very popular because of their good efficiency; and are used in most computer programs for differential equations. They are single-step methods, as the Euler methods.

19. Runge-Kutta Methods To convey some idea of how the Runge-Kutta is developed, let’s look at the derivation of the 2nd order. Two estimates

20. Runge-Kutta Methods The initial conditions are: The Taylor series expansion

21. Runge-Kutta Methods From the Runge-Kutta The definition of the function Expand the next step

22. Runge-Kutta Methods From the Runge-Kutta Compare with the Taylor series 4 Unknowns

23. Runge-Kutta Methods The Taylor series coefficients (3 equations/4 unknowns) If you select “a” as If you select “a” as Note: These coefficient would result in a modified Euler or Midpoint Method

24. Runge-Kutta Method (2nd Order) Example Consider Exact Solution The initial condition is: The step size is: Use the coefficients

25. Runge-Kutta Method (2nd Order) Example The values are

26. Runge-Kutta Method (2nd Order) Example The values are equivalent of Modified Euler

27. Runge-Kutta Method (2nd Order) Example [b] The values are

28. Runge-Kutta Method (2nd Order) Example [b] The values are

29. Runge-Kutta Methods The Runge-Kutta methods are higher order approximation of the basic forward integration. These methods provide solutions which are comparable in accuracy to Taylor series solution in which higher order derivatives are retained. It should be noted that the equations are not need to be linear.

30. Runge-Kutta Methods

31. The 4th Order Runge-Kutta The general form of the equations:

32. The 4th Order Runge-Kutta This is a fourth order function that solves an initial value problems using a four step program to get an estimate of the Taylor series through the fourth order. This will result in a local error of O(Dh5) and a global error of O(Dh4)

33. 4th-orderRunge-Kutta Method f2 f4 f3 f1 xi xi + h/2 xi + h

34. Runge-Kutta Method (4th Order) Example Consider Exact Solution The initial condition is: The step size is:

35. The 4th Order Runge-Kutta The example of a single step:

36. Runge-Kutta Method (4th Order) Example The values for the 4th order Runge-Kutta method

37. Runge-Kutta Method (4th Order) Example The values are equivalent to those of the exact solution. If we were to go out to x=5. y(5) = -111.4129 (-111.4132) The error is small relative to the exact solution.

38. Runge-Kutta Method (4th Order) Example A comparison between the 2nd order and the 4th order Runge-Kutta methods show a slight difference.

39. The 4th Order Runge-Kutta Higher order differential equations can be treated as if they were a set of first-order equations. Runge-Kutta type forward integration solutions can be obtain. A more direct solution can be obtained by repeating the whole process used in first-order cases.

40. The 4th Order Runge-Kutta The general form of the 2nd order equations:

41. The step sizes are: The next step would be:

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