We can use taylors theorem to approximate derivatives

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We can use Taylor’s theorem to approximate derivatives of a function, given only the function’s value on a uniform grid. For example, consider the following corollary. Corollary 7.3 (Forward Difference) . Suppose that u ( x ) : [ a, b ] R is smooth (infinitely differentiable) function, and we have approximated this on a uniform grid by { u j | j = School of Mathematics 143 University of Manchester
MATH20401: 7 NUMERICAL SOLUTION OF PDES S.L. COTTER 0 , 1 , . . . , N } at the points { x j = a + j × h | j = 0 , 1 , . . . , N } , where h = ( b - a ) /N . Then if x = a + Jh u J +1 - u J h = u 0 ( x ) + O ( h ) . Proof. By Taylor’s theorem (with n = 2), we have that there exists ξ [ a, b ] such that u ( x + h ) = u ( x ) + hu 0 ( x ) + h 2 2 u 00 ( ξ ) = u ( x ) + hu 0 ( x ) + O ( h 2 ) . Since u ( x + h ) = u (( J + 1) h ) = u J +1 and u ( x ) = u ( Jh ) = u J , we therefore have that: u J +1 - u J h = hu 0 ( x ) + O ( h 2 ) h , = u 0 ( x ) + O ( h ) . Note here that if we divide an O ( h n ) function by h m where m < n , then we get an O ( h n - m ) function. Proof: Exercise. This result means that if h is small enough, then we can use ( u J +1 - u J ) /h as an approximation for u ( x ). This particular method is a first order method of approximation of the first derivative. Definition 7.4. A finite difference method is said to be an n - th order approximation if the remainder term is O ( h n ) . The remainder term describes how big an error we are making in our approximation of the function that we want to understand, and since h 1, then h n is increasingly small as n gets larger. Therefore higher order methods are more accurate (the error in the ap- proximation is smaller). However, usually higher order methods are more computationally expensive , i.e. they require the computer to calculate more operations. In all of numerical analysis and scientific computing, there is always a trade off between computation cost (the number of operations that the computer must make) and the accuracy of the algorithm (the size of the error in the approximation of the function). Let us now look at another finite difference approximation of the first derivative. 7.1.1 Centered Differencing Corollary 7.5 (Centered Differencing) . Suppose that u ( x ) : [ a, b ] R is a smooth func- tion, and we have approximated u ( x ) on a uniform grid by N + 1 points. Let us define δu ( x ) = u ( x + h ) - u ( x - h ) = u J +1 - u J - 1 (assuming that x = a + Jh for some J ). Then δu ( x ) 2 h is an second order approximation for u 0 ( x ) . i.e., δu ( x ) 2 h = u 0 ( x ) + O ( h 2 ) . School of Mathematics 144 University of Manchester
MATH20401: 7 NUMERICAL SOLUTION OF PDES S.L. COTTER Proof. We can look again to the Taylor’s expansions for both u ( x + h ) (as in the proof of the last corollary), and u ( x - h ). Note that the Taylor expansion for u ( x - h ) is found by replacing h with - h in the formula: u ( x - h ) = u ( x ) - hu 0 ( x ) + h 2 2 u 00 ( x ) + O ( h 3 ) . Therefore δ 2 h u ( x ) = u ( x + h ) - u ( x - h ) 2 h , = ( u ( x ) + hu 0 ( x ) + h 2 u 00 ( x ) / 2) - ( u ( x ) - hu 0 ( x ) + h 2 u 00 ( x ) / 2) + O ( h 3 ) 2 h , = 2 hu 0 ( x ) + O ( h 3 ) 2 h , = u 0 ( x ) + O ( h 2 ) . Therefore δu ( x ) 2 h is a second order approximation of the first derivative of u .

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