CFD lecture 2 - 1 time index du u n +1 u n = + ( t ) dt t n...

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P () time index 1 nn n du u u t dt t σ + =+ 1st order accuracy in time Leap - Frog difference (central difference) 11 2 2 du u u t dt t +− 2 nd order accuracy in time Many higher order schemes with different properties have been developed, we will revisit in detail later. The method for determining the order of accuracy is by expanding in Taylor Series. ( ) ( ) ( ) ( ) 23 1 000 0 0 1 0 0 : .... 2! 3! .... 3! n n tt uuu uu u t t t t t tt t u t t t + ∆∆ ∂∂ ∂ +∆ = + + + −∆ = + + With. 1 0 n ut t u + = 1 0 n t u = Substitute into expression for Leap - Frog (L-F) () ( ) 3 3 0 5 0 3 2 3 0 3 2 2 1 ... 22 2 3 ! 26 nn u t u t t t t u u t t −∂ ∆∂ + H.O.T. higher order terms. With “small” say t 1 100 t ∆= in some normalized sense then 2 t “truncation error” terms are 1 part in 10,000 which is generally sufficiently accurate for 2
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engineering approximations. None the less, 3 3 1 6 u t may be large locally and errors may be large.
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This note was uploaded on 06/08/2011 for the course EOC 6850 taught by Professor Slinn during the Spring '07 term at University of Florida.

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CFD lecture 2 - 1 time index du u n +1 u n = + ( t ) dt t n...

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