Finite Difference Calculus

Transcription

1 Chapter 3 Discretization of Computational Domain An Introduction to First Session Contents: 1) Approximation of Derivatives 2) Order Symbols 3) High-Order Derivatives 4) Richardson s Extrapolation (The deferred approach to the limit) 1 2 Brook Taylor Born: August 18, 1685 Municipal Borough of Edmonton Died: November 30, 1731 London, United Kingdom Education: St John's College, Cambridge, University of Cambridge Brook Taylor w an English mathematician who is best known for Taylor's theorem and the Taylor series If a function f(x) is infinitely differentiable at x = x 0, we can express: To find the coefficients, initially, we put x = x 0 : Taking the first derivative gives: 3 4 1

2 Similarly, we take derivative again: Now, we can write for other functions. since Putting x = x 0 results in: In this way, we may conclude: we have: where Eventually, we can write of f(x) at x = x 0 : As of sin(x) and cos(x) at x = x Also, we already know the its is: Order Symbols Instead of saying that sin(x) tends to zero at the same rate that x tends to zero, we say: Big Oh In general: To give another example, recall that: If and Hence: 7 8 2

3 Order Symbols For example: Let s write for f(x + h) at x since Other examples for x 0 Finite Difference Collecting all terms of O h : Forward Difference Let s re-write bed on index notation: 9 10 Defining Forward Difference Operator: Similarly for f(x h) at P we have We have: NOTE: Finite Difference Truncation error is the difference between the derivative and its finite difference Approximation. Collecting all terms of O h : For the Forward Difference : Backward Difference Let s re-write bed on index notation: Limited

4 Defining Backward Difference Operator: Let s re-write both at P We have: We may write f (x) explicitely: Finite Difference Higher-Order Derivatives Let s take a look at these two Taylor Series: Collecting all terms of O h 2 : Finite Difference ( 2) Central Difference we have: Bed on index notation we have: Solving for f (x) yields: or

5 Higher-Order Derivatives Recall: Higher-Order Forward and Backward Difference Recall: Solving for f x gives: Therefore, we can write forward difference for f (x) in operator notation Simplifying of this equation results in: Similarly, backward difference for f (x) in operator notation would be: or: Finite Difference Discretization Forward Difference O(h) Finite Difference Discretization Forward Difference O(h) Central Difference O(h 2 ) Central Difference O(h 4 ) Backward Difference O(h) Backward Difference O(h)

6 Example1 If f x = e x find f (1) using forward difference, choosing h = 0.1 Example1 Now, let s choose h = 0.05 Forward Difference Using Central Difference: Slope of line in f - h 2 coordinate system is: Exact value at x = 1 Central Difference Exact Forward Difference Central Difference Value Relative Error % 0.17% we can find exact value of f x = 1 : Example2 If f x = sin(x) find f (1) using central difference, choosing h = 0.2 Example2 The exact answer is: Using the method introduced in previous example: Let s choose h = 0.1 Which gives: Let s choose h =

7 Example3 Example3 Consider f x = sin(10πx), find f 0 by choosing h = 0.2 The exact answer is: Consider f x = sin(10πx), find f 0 by choosing h = 0.2 The exact answer is: Problem? T = h Forward Difference Forward Difference Central Difference Central Difference Richardson Extrapolation Richardson Extrapolation Lewis Fry Richardson Born: October 11, 1881 Newctle upon Tyne, United Kingdom Died: September 30, 1953 Education: Kilmun, United Kingdom Bootham School, Newctle University, King's College, Cambridge, University of London, Durham University Analytical Solution Numerical Solution T.E. is constant He w an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecting. Question Is it possible to obtain the exact solution from numerical solution? x o x 0 +h x 0 +h/2 x 0 +2h y n x 0 +nh y 2n 27 x o x 0 +2(h/2) x 0 +2n(h/2) 28 7

8 Richardson Extrapolation Richardson Extrapolation Second order approximation Regardless of terms, C can be obtained By Replacing, C in we have Higher order approximation Consequently, the error for step size of h/2 can be determined