Runge_Kutta.ppt

# Heuns method x y x i x i1 y i1 predicted y i figure 1

• No School
• AA 1
• 20

This preview shows page 7 - 12 out of 20 pages.

Heun’s Method x y x i x i+1 y i+1 , predicted y i Figure 1 Runge-Kutta 2nd order method (Heun’s method) h k y h x f Slope i i 1 , i i i i y x f h k y h x f Slope Average , , 2 1 1 i i y x f Slope , Heun’s method 2 1 1 a 1 1 p 1 11 q resulting in h k k y y i i 2 1 1 2 1 2 1 where i i y x f k , 1 h k y h x f k i i 1 2 , Here a 2 =1/2 is chosen

Subscribe to view the full document.

Midpoint Method Here 1 2 a is chosen, giving 0 1 a 2 1 1 p 2 1 11 q resulting in h k y y i i 2 1 where i i y x f k , 1 h k y h x f k i i 1 2 2 1 , 2 1
Ralston’s Method Here 3 2 2 a is chosen, giving 3 1 1 a 4 3 1 p 4 3 11 q resulting in h k k y y i i 2 1 1 3 2 3 1 where i i y x f k , 1 h k y h x f k i i 1 2 4 3 , 4 3

Subscribe to view the full document.

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 K dt d 1200 0 , 10 81 10 2067 . 2 8 4 12 Find the temperature at 480 t seconds using Heun’s method. Assume a step size of 240 h seconds. 8 4 12 10 81 10 2067 . 2 dt d 8 4 12 10 81 10 2067 . 2 , t f h k k i i 2 1 1 2 1 2 1
Solution Step 1: K t i 1200 ) 0 ( , 0 , 0 0 0

Subscribe to view the full document.

• Fall '19
• Trigraph, Yi, Partial differential equation, y 1.3e  x

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern