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CFD lecture 2

CFD lecture 2 - 1 time index du u n 1 u n = t dt t n 1st...

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P ( ) time index 1 n n n du u u t dt t σ + = + 1st order accuracy in time Leap - Frog difference (central difference) ( ) 1 1 2 2 n n n du u u t dt t σ + = + 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. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 2 3 1 0 0 0 0 0 2 3 2 3 2 3 1 0 0 0 0 0 2 3 : .... 2! 3! : .... 2! 3! n n t t u u u u u t t u t t t t t t t t t t u u u u u t t u t t t t t t t t + + ∆ = + ∆ + + + − ∆ = − ∆ + + With. ( ) 1 0 n u t t u + + ∆ = ( ) 1 0 n u t t u − ∆ = Substitute into expression for Leap - Frog (L-F) ( ) ( ) ( ) ( ) ( ) ( ) 3 1 1 3 0 5 0 3 2 1 1 3 0 3 2 2 1 ... 2 2 2 3! 2 6 n n n n u t t t u u u t t t t t t t t u u u u t t t t σ + + = + = + + 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
engineering approximations.

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CFD lecture 2 - 1 time index du u n 1 u n = t dt t n 1st...

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