CSC_349_HANDOUT#36

# CSC_349_HANDOUT#36 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 36 RUNGE-KUTTA METHODS (Section 25.3, page 701 in the 5 th ed. or page 727 in the 6 th ed.) Advantage of Taylor methods of order n global truncation error of ) ( n h O insures high accuracy (even for n = 3, 4 or 5) Disadvantage high order derivatives of )) ( , ( x y x f may be difficult and expensive to evaluate. Runge-Kutta methods are higher order formulas (they can have any order 1 ) that require function evaluations only of )) ( , ( x y x f , and not of any of its derivatives. This is accomplished using the Taylor polynomial for a function of 2 variables: ) , ( ) , ( ) , ( ) , ( y x f k y x f h y x f k y h x f y x + + = + + L + + + + + + + + ) , ( 6 ) , ( 2 ) , ( 2 ) , ( 6 ) , ( 2 ) , ( ) , ( 2 3 2 2 3 2 2 y x f k y x f hk y x f k h y x f h y x f k y x f hk y x f h yyy xyy xxy xxx yy xy xx where y x f f x f f xy x 2 , , etc. The derivation of Runge-Kutta methods and an understanding of why they work requires the Taylor polynomial for a function of 2 variables, but this Taylor polynomial is not required to use these methods to numerically approximate the solution of a differential equation. A second order Runge-Kutta formula is derived in the textbook on page 703; however, we will not consider their derivation, only their form and how to use them. Runge-Kutta methods are so-called one-step methods (as also are Euler’s method and all Taylor methods): that is, they are of the form (see (25.28) on p. 701 in 5 th ed. or p. 727 in 6 th ed.) ) , , ( 1 h y x h y y i i i i Φ + = +

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2 for some (possibly very complicated) function Φ . That is, each computed approximation 1 + i y is computed using only the value i y at the previous grid point, along with the values
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CSC_349_HANDOUT#36 - COMPUTER SCIENCE 349A Handout Number...

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