DERIVATION OF DIFFERENCE APPROXIMATIONS USING UNDETERMINED COEFFICIENTS

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Mark Conrad Daniels
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1 LECTURE 7 DERIVATION OF DIFFERENCE APPROXIMATIONS USING UNDETERMINED COEFFICIENTS All discrete approximations to derivatives are linear combinations of functional values at the nodes a f a f a f E p h p The total number of nodes used must be at least one greater than the order of differentiation p to achieve minimum accuracy Oh. To obtain better accuracy, you must increase the number of nodes considered. For central difference approximations to even derivatives, a cancelation of truncation error terms leads to one order of accuracy improvement p. 7.1

2 Forward second order accurate approximation to the first derivative 1 Develop a forward difference formula for which is E Oh accurate First derivative with Oh accuracy the minimum number of nodes is First derivative with Oh accuracy need 3 nodes i i1 i The first forward derivative can therefore be approximated to Oh as: df dx x T.S. expansions about are: x i x i E h 1 3 h ----f i ----f 6 i 1 h h 3 Oh 4 1 h h --h 3 3 f 3 i 4 Oh 4 p. 7.

3 1 Substituting into our assumed form of and re-arranging f h h i Desire and nd order accuracy coefficient of must equal unity and coefficients of and must vanish h hfi h fi 1 Oh h 0 p. 7.3

4 Solving these simultaneous equations 1 3,, Thus the equation now becomes 3 -- f i 1 -- f i f h i h fi 6 3 Oh h f 3 h 3 Oh 3 The forward difference approximation of nd order accuracy E where E h 1 --h 3 f 3 i p. 7.4

5 Forward first order accurate approximation to the second derivative Derive the Oh forward difference approximations to Second derivative 3 nodes for Oh accuracy E h Develop Taylor series expansions for, 1 and, substitute into expression and re-arrange: h h f h i h 6 --f 3 i Oh p. 7.5

6 In order to compute we must have: h 1 -- h ,, 3 1 Therefore 1 h E where E h p. 7.6

7 Skewed fourth order accurate approximation to the second derivative Develop a fourth order accurate approximation to the second derivative at node i which involves nodes i 1, i and subsequent nodes to the right of node i requires 3 nodes for Oh accuracy requires 4 nodes for Oh accuracy requires 5 nodes for Oh 3 accuracy requires 6 nodes for Oh 4 accuracy Therefore we consider nodes i-1 i i1 i i3 i4 is approximated as: E h p. 7.7

8 Steps to solve for the unknown coefficients in the linear combination for Develop Taylor series expansions for,,,, Substitute and re-arrange to collect terms on equal derivatives Generate equations by setting coefficients of to 1 and the remaining 5 leading coefficients to zero p. 7.8

9 NUMERICAL DIFFERENTIATION USING DIFFERENCE OPERATORS Difference Operators First order difference operators Consider the following full and intermediate nodes h h i-1 x i-1/ i x i1/ i1 First order forward difference operator 1 First order backward difference operator 1 First order central difference operator defined using full node functional values 1 1 p. 7.9

10 Notes Intermediate functional values are defined as f 1 i -- f h x i -- First order central difference operator defined using intermediate nodes f 1 i -- f 1 i -- The central difference operator is defined at an intermediate node as f 1 i -- 1 The order of the difference operator is related to the number of times that the operator is applied and not to the order of accuracy Higher order difference operators simply repeat operation as indicated by the operator p. 7.10

11 Second order forward difference operator f i p. 7.11

12 Third order backward difference operator 3 1 f i 1 f 1 i p. 7.1

13 Second order central difference operator f i f 1 i -- f 1 i -- f 1 i -- f 1 i p. 7.13

14 Second order mixed difference operator We can also apply different operators; e.g. n m m, 1 m n Applying a first order forward difference operator and then a first order backward difference operator f i We note that and in general m m m (m) th order central difference operator p. 7.14

15 Approximations to Differentiation Using Difference Operators df dx df dx df dx x i x i x i x i x i x i First order backward difference operator approximation to the first derivative x i x i h i-1 i h 1 p. 7.15

16 First order central difference operator approximation to the first derivative x i h/ h/ i-1/ i i1/ x i h 1 f 1 f 1 i -- i h h h i-1/ i i1/ x i h h p. 7.16

17 Central difference approximation to the first derivative as an average of first order forward and backward difference approximations We note that first order central difference approximations can also be derived as arithmetic averages of first order forward and backward difference approximations x i x i h h h This concept can be generalized to central approximations to higher order derivatives as well (see the next section) p. 7.17

18 General difference operator approximations to derivatives In general we can approximate derivatives using Forward approximations Backward approximations p p h p Oh Central approximations p p p h p Oh p f p p f p i -- i h p Oh p even p p f p 1 p f p 1 i i h p Oh p odd p. 7.18

19 A complete operator approach to central differencing can be developed. However this approach is somewhat artificial and overly complicated. p. 7.19

5 Numerical Differentiation

D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives