lecture 13

lecture 13 - CE 30125 - Lecture 13 LECTURE 13 NUMERICAL...

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CE 30125 - Lecture 13 p. 13.1 LECTURE 13 NUMERICAL DIFFERENTIATION FORMULAE BY INTERPOLATING POLY- NOMIALS Relationship Between Polynomials and Finite Difference Derivative Approximations • We noted that N th degree accurate Finite Difference (FD) expressions for first derivatives have an associated error • If f(x) is an N th degree polynomial then, and the FD approximation to the first derivative is exact! • Thus if we know that a FD approximation to a polynomial function is exact, we can derive the form of that polynomial by integrating the previous equation. Eh N d N 1 + f dx N 1 + --------------- d N 1 + f N 1 + ---------------- 0 = fx  a 1 x N a 2 x N 1 a N 1 + ++ +
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CE 30125 - Lecture 13 p. 13.2 • This implies that a distinct relationship exists between polynomials and FD expressions for derivatives (different relationships for higher order derivatives). • We can in fact develop FD approximations from interpolating polynomials Developing Finite Difference Formulae by Differentiating Interpolating Polynomials Concept • The approximation for the derivative of some function can be found by taking the derivative of a polynomial approximation, , of the function . Procedure • Establish a polynomial approximation of degree such that is forced to be exactly equal to the functional value at data points or nodes • The derivative of the polynomial is an approximation to the derivative of p th fx  p gx N Np N 1 + p p
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CE 30125 - Lecture 13 p. 13.3 Approximations to First and Second Derivatives Using Quadratic Interpolation • We will illustrate the use of interpolation to derive FD approximations to first and second derivatives using a 3 node quadratic interpolation function • For first derivatives p= 1 and we must establish at least an interpolating polynomial of degree N= 1 with N+ 1=2 nodes • For second derivatives p= 2 and we must establish at least an interpolating polyno- mial of degree N= 2 with N+ 1=3 nodes • Thus a quadratic interpolating function will allow us to establish both first and second derivative approximations • Apply a shifted coordinate system to simplify the derivation without affecting the gener- ality of the derivation h h x 0 x 1 x 2 f 0 f 1 f 2 x shifted x axis h 2 h 0
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CE 30125 - Lecture 13 p. 13.4 Develop a quadratic interpolating polynomial • We apply the Power Series method to derive the appropriate interpolating polynomial • Alternatively we could use either Lagrange basis functions or Newton forward or backward interpolation approaches in order to establish the interpolating polyno- mial • The 3 node quadratic interpolating polynomial has the form • The approximating Lagrange polynomial must match the functional values at all data points or nodes ( , , ) gx  a o x 2 a 1 xa 2 ++ = N 1 + x o 0 = x 1 h = x 2 2 h = o f o = a o 0 2 a 1 0 a 2 f o = 1 f 1 = a o h 2 a 1 ha 2 f 1 = 2 f 2 = a o 2 h 2 a 1 2 h a 2 f 2 =
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CE 30125 - Lecture 13 p. 13.5 • Setting up the constraints as a system of simultaneous equations • Solve for , , , , • The interpolating polynomial and its derivative are equal to: 0 1 h 2 h 1 4 h 2 2 h 1 a o a
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lecture 13 - CE 30125 - Lecture 13 LECTURE 13 NUMERICAL...

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