Lecture 13 - Example First Derivatives Use forward and...

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Unformatted text preview: Example: First Derivatives ❧ Use forward and backward difference approximations to estimate the first derivative of ❧ at x = 0.5 with h = 0.5 and 0.25 (exact sol. = f ′ = -0.9125) ❧ Forward Difference ❧ Backward Difference 2 . 1 x 25 . x 5 . x 15 . x 1 . ) x ( f 2 3 4 +---- = =- =- =-- = ′ = =- =- =-- = ′ = % . , . . . . . . ) . ( ) . ( ) . ( , . % . , . . . . . ) . ( ) ( ) . ( , . 5 26 155 1 25 925 63632813 5 75 5 f 75 f 5 f 25 h 9 58 45 1 5 925 2 5 1 5 f 1 f 5 f 5 h t t ε ε =- =- =-- = ′ = =- =- =-- = ′ = % . , . . . . . . ) . ( ) . ( ) . ( , . % . , . . . . . ) ( ) . ( ) . ( , . 7 21 714 25 10351563 1 925 25 5 25 f 5 f 5 f 25 h 7 39 55 5 2 1 925 5 f 5 f 5 f 5 h t t ε ε Example: First Derivative ❧ Use central difference approximation to estimate the first derivative of at x = 0.5 with h = 0.5 and 0.25 (exact sol. = f ′ = -0.9125) ❧ Central Difference 2 . 1 x 25 . x 5 . x 15 . x 1 . ) x ( f 2 3 4 +---- = % 4 . 2 , 934 . 5 . 10351563 . 1 63632813 . 25 . 75 . ) 25 . ( f ) 75 . ( f ) 5 . ( f , 25 . h % 6 . 9 , . 1 1 2 . 1 2 . 1 ) ( f ) 1 ( f ) 5 . ( f , 5 . h =- =- =-- = ′ = =- =- =-- = ′ = t t ε ε Second-Derivatives ❧ Taylor-series expansion ❧ Uniform grid spacing ❧ Second-order accurate O(h 2 )...
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Lecture 13 - Example First Derivatives Use forward and...

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