Applying a first order forward difference operator

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, Applying a first order forward difference operator and then a first order backward differ- ence operator • We note that and in general ( 2m ) th order central differ- ence operator n m Δ m 1 m n ∇Δ f i Δ ∇ f i ( ) = Δ f i f i 1 ( ) = Δ f i Δ f i 1 = f i 1 + f i ( ) f i f i 1 ( ) = Δ∇ f i f i 1 + 2 f i f i 1 + = δ 2 Δ∇ ∇Δ = = δ 2 m m Δ m =
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CE 30125 - Lecture 12 p. 12.15 Approximations to Differentiation Using Difference Operators First order backward difference operator approximation to the first derivative df dx ----- x i Δ f i Δ x i ------- df dx ----- x i f i x i -------- df dx ----- x i δ f i δ x i ------- f i 1 ( ) f i x i -------- i-1 i ° x i = h f i 1 ( ) f i f i 1 h -------------------
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CE 30125 - Lecture 12 p. 12.16 First order central difference operator approximation to the first derivative f i 1 ( ) δ f i δ x i ------- i-1/2 i i+1/2 h/2 h/2 d x i = h f i 1 ( ) f i 1 2 -- + f i 1 2 -- h -------------------------- i-1/2 i i+1/2 h h d x i = 2h f i 1 ( ) f i 1 + f i 1 2 h --------------------------
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CE 30125 - Lecture 12 p. 12.17 Central difference approximation to the first derivative as an average of first order for- ward and backward difference approximations We note that first order central difference approximations can also be derived as arith- metic averages of first order forward and backward difference approximations This concept can be generalized to central approximations to higher order derivatives as well (see the next section) f i 1 ( ) 1 2 -- Δ f i Δ x i ------- f i x i -------- + f i 1 ( ) 1 2 -- f i 1 + f i h ------------------ f i f i 1 h ------------------ + f i 1 ( ) f i 1 + f i 1 2 h -------------------------- =
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CE 30125 - Lecture 12 p. 12.18 General difference operator approximations to derivatives In general we can approximate derivatives using • Forward approximations • Backward approximations • Central approximations even odd f i p ( ) Δ p f i h p --------- O h ( ) + f i p ( ) p f i h p ---------- O h ( ) + f i p ( ) p f i p 2 -- + Δ p f i p 2 -- + 2 h p ---------------------------------------- O h ( ) 2 + p f i p ( ) p f i p 1 2 ----------- + Δ p f i p 1 2 ----------- + 2 h p ------------------------------------------------------ O h ( ) 2 + p
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CE 30125 - Lecture 12 p. 12.19
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  • Fall '08
  • Westerink,J
  • Numerical Analysis, Mathematical analysis, finite difference, Numerical Differentiation

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