over The idea is to construct an algorithm to solve the IVP ODE (8.1)

Transcription

1 Runge- Ku(a Methods

2 Review of Heun s Method (Deriva:on from Integra:on) The idea is to construct an algorithm to solve the IVP ODE (8.1) over To obtain the solution point we can use the fundamental theorem of calculus and integrate to get over (8.2)

3 Review of Heun s Method (Deriva:on from Integra:on) When equation (8.2) is solved from the result is (8.3) A numerical integration method can be used to approximate the definite integral in equation (8.3). If the trapezoidal rule (Lecture 5) is used with the step size the result is (8.4)

4 Review of Heun s Method (Deriva:on from Integra:on) Notice that the formula on the right hand side of (8.4) involves the yet to be determined value To estimate we use Euler s forward method or, for our case (8.5) Substituting (8.5) into (8.4), the resulting formula is called Heun s method, given as (8.6)

5 Review of Heun s Method (Deriva:on from Integra:on) In Heun s method, at each step: (a) the Euler s (forward) method is used as a prediction, and then, (b) the trapezoidal rule is used to make a correction to obtain the final value. The general step for Heun s method is Predictor: Corrector: (8.7)

6 Example Use Heun s method to solve the IVP Compare solutions for Solution: (1) For we use to calculate

7 Example, the predictor: which we use to calculate Thus, the corrector:

8 Example x i y i y exact h = 0.5 h = 0.25 h = (0.0127%) (0.1055%) (0.0252%) (0.037%) (0.8785%) (0.1999%) (0.0477%) (0.2726%) (0.0649%) (1.3986%) (0.3177%) (0.0758%) (1.4623%) (0.3318%) (0.0791%) (1.2639%) (0.2865%) (0.0683%) (1.0004%) (0.2265%) (0.054%) (0.7626%) (0.1725%) (0.0411%)

9 MATLAB Implementa:ons function [x,y] = HeunMethod(f,xinit,xend,yinit,h) % Number of iterations N = (xend-xinit)/h; x = zeros(1, N); y = zeros(1, N); y(1) = yinit; x(1) = xinit; for i=1:n x(i+1) = x(i)+h; y0 = y(i) + feval(f,x(i),y(i))*h; pred1 = feval(f,x(i),y(i)); pred2 = feval(f,x(i+1),y0); y(i+1) = y(i) + h*(pred1+pred2)/2; end function dydx = Example_HeunMethod(x,y) dydx = (x-y)/2; end

10 MATLAB Implementa:ons

11 Runge- Ku(a Method Here we see that Heun's method is an improvement over the rather simple Euler s (forward) method. Although the method uses Euler's method as a basis, it goes beyond it, it attempts to compensate for the Euler method's failure to take the curvature of the solution curve into account. Heun's method is one of the simplest of a class of methods called predictor-corrector algorithms. One of the most powerful predictor-corrector algorithms of all one which is so accurate, that most computer packages designed to find numerical solutions for differential equations will use it by default is the fourth order Runge-Kutta method.

12 Second- Order Runge- Ku(a Methods The 2 nd order Runge-Kutta method simulates the accuracy of the Taylor series method of order 2. Recall the Taylor series formula for Where C T is a constant involving the third derivative of and the other terms in the series involve powers of for n > 3. (8.8) The derivatives must expressed in terms of and its partial derivatives. Recall that (8.9)

13 Second- Order Runge- Ku(a Methods Equation (8.9) can be differentiated using the chain rule for differentiating a function of two variables, for example: then Back to the equation (8.9),

14 Second- Order Runge- Ku(a Methods Another notation for partial derivatives: hence (8.10) Substitute the derivatives (8.9) and (8.10) into the Taylor series (8.8): (8.11)

15 Second- Order Runge- Ku(a Methods For second order, equation (8.11) can be rewritten as a linear combination of two function values to express where (8.12) (8.13) Next, we determine the values of A, B, P and Q.

16 Second- Order Runge- Ku(a Methods Recall Taylor series for functions more than one variable (Lecture 1): and use it to expand f 1 in equation (8.13) to get (8.14)

17 Second- Order Runge- Ku(a Methods Then (8.14) is used in (8.12) to get the second-order Runge- Kutta expression for or (8.15)

18 Second- Order Runge- Ku(a Methods Comparing similar terms in equations (8.11) and (8.15): implies that implies that implies that

19 Second- Order Runge- Ku(a Methods Hence, we require that A, B, P, and Q satisfy the relations (8.16) The second-order Runge-Kutta method in (8.15) will have the same order of accuracy as the Taylor s method in (8.11). Now, there are 4 unknowns with only three equations, hence the system of equations (8.16) is undetermined, and we are permitted to choose one of the coefficients. Case (i): Choose This choice leads to

20 Second- Order Runge- Ku(a Methods Substituting these values into equation (8.12) and (8.13) yields (8.17) Case (ii): Choose This choice leads to Substituting these values into equation (8.12) and (8.13) yields (8.18)

21 Example MATLAB implementations of 2 nd order Runge-Kutta methods for IVP

22 Example MATLAB implementations of 2 nd order Runge-Kutta methods for IVP

23 Third- Order Runge- Ku(a Methods The general formula is where (8.19)

24 Fourth- Order Runge- Ku(a Methods The classical fourth-order Runge-Kutta method where (8.20)

25 Higher- Order Runge- Ku(a Methods (1) Butcher s fifth-order Runge-Kutta method: where (8.21)

26 Higher- Order Runge- Ku(a Methods (1) Butcher s fifth-order Runge-Kutta method:

27 Higher- Order Runge- Ku(a Methods (2) Runge-Kutta-Fehlberg (RKF45) method Each step requires the use of the following six values

28 Higher- Order Runge- Ku(a Methods An approximation using a fourth-order Runge-Kutta method is given by: (8.22)

29 Example MATLAB implementations of 2 nd order, 3 rd order, and 4 th order Runge-Kutta methods for IVP and step size The exact solution is

30 h = 0.2

31 h = 0.1

32 Precision of the Runge- Ku(a Method Assume that y(x) is the solution of the IVP. If is the sequence of approximations generated by the Runge-Kutta method of order 4, then In particular, the global error at the end of the interval will satisfy

33 Precision of the Runge- Ku(a Method If approximations are computed using the step size h and h/2, we should have The idea is that, if the step size in Runge-Kutta order 4 is reduced by a factor of ½, we can expect that the global error will be reduced by a factor of