lecture 21

# 1 ce 30125 lecture 21 p 2111 insert figure no 99

• Notes
• 13

This preview shows page 10 - 13 out of 13 pages.

1 + , ( ) + + + =

Subscribe to view the full document.

CE 30125 - Lecture 21 p. 21.11 INSERT FIGURE NO. 99 • Notes • Find the point by using the weighted average of the 4 slopes • Note that there are other coefficients possible for 4th order Runge-Kutta • We require 4 evaluations of the slope for this 4th order method 4 times the work of the 1st order Runge-Kutta Method y y j+1/2 t j+1 = t j t y j t j + D t 2 t j + D t y j+1/2 * * * y j+1 * * use further improved value of slope to obtain y j +1 3 f(y j + 1/2 , t j + 1/2 ) ** 3 use improved slope to evaluate the new midpoint y j +1/2 ** 2 f(y j + 1/2 , t j + 1/2 ) * 2 f(t j + y j ) used to estimate y j+1/2 * 1 , y j 1 +
CE 30125 - Lecture 21 p. 21.12 Summary of Runge-Kutta Methods Self starting and interval can be changed at any time without complications Very easy to program Comparable if not better accuracy than other methods Require much more computer time than other methods of comparable accuracy since • Number of functional evaluations is proportional to the accuracy • Functional evaluations can not be reused Local truncation errors are difficult and expensive to obtain (easier for other methods) Δ t

Subscribe to view the full document.

CE 30125 - Lecture 21 p. 21.13 Qualitative basis for verifying accuracy of solutions use 2 different time steps (similar to Romberg integration) • Can estimate truncation error as: where solution found using (therefore 2 steps) solution found using (1 step) = order of the method • Need to run the solution using two different time steps!
You've reached the end of this preview.
• Fall '08
• Westerink,J
• dt, Euler, Runge–Kutta methods, t j, INSERT FIGURE NO.

{[ 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