D 2 y dt 2 j o δ t 3 dy dt f t y d 2 y dt 2 f t f y

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d 2 y dt 2 -------- j O Δ t ( ) 3 + + + = dy dt ----- f t y , ( ) = d 2 y dt 2 -------- f t ---- f y ----- dy dt ----- + = d 2 y dt 2 -------- f t ---- f y ----- f + =
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CE 30125 - Lecture 21 p. 21.5 Substituting for and into the Taylor series expanded form of , Equation (5), we obtain an exact expression for (to ) (6) Recall that the Runge-Kutta approximation for was: Select constants between Equations and (6) such that both are the same dy dt ----- j d 2 y dt 2 -------- j y j 1 + y j 1 + O Δ t ( ) 3 y j 1 + y j Δ t f j Δ t ( ) 2 2 ------------ f t ---- j f y ----- j f j + O Δ t ( ) 3 + + + = y j 1 + y j 1 + y j Δ t a 1 a 2 + ( ) f j Δ t ( ) 2 a 2 p 1 f t ---- j a 2 p 2 f j f y ----- j + O Δ t ( ) 3 + + + = 4' ( ) 4' ( ) a 1 a 2 + 1 = a 2 p 1 1 2 -- = a 2 p 2 1 2 -- = a 1 1 a 2 = p 1 p 2 1 2 a 2 -------- = =
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CE 30125 - Lecture 21 p. 21.6 4 unknowns and 3 equations Solve in terms of an arbitrary constant The local or per time step error is These methods will be accurate Improved Euler Method (Modified Euler-Cauchy) • Let Substituting values into the standard form of the Runge-Kutta algorithm, Equations (1) and (3): a a 1 1 a = a 2 a = p 1 p 2 1 2 a ----- = = O Δ t ( ) 3 O Δ t ( ) 2 a 1 2 -- = a 1 a 2 1 2 -- = = p 1 p 2 1 = = y j 1 + y j 1 2 -- Δ t f j 1 2 -- Δ t f t j Δ t y j Δ t f j + , + ( ) + + =
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CE 30125 - Lecture 21 p. 21.7 INSERT FIGURE NO. 97 • Procedure • Estimate by using first order Euler • Evaluate the slope at and the 1st order estimate of • Find the point by taking the average of slope at and at ( , 1st order
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  • Fall '08
  • Westerink,J
  • dt, Euler, Runge–Kutta methods, t j, INSERT FIGURE NO.

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