CE 30125 - Lecture 8 LECTURE 8 NUMERICAL DIFFERENTIATION FORMULAE BY INTERPOLATING POLY- NOMIALS Relationship Between Polynomials and Finite Difference Derivative Approximations • We noted that Nth degree accurate Finite Difference (FD) expressions for first derivatives have an associated error N + 1 N d f Eh ---------------- dxN + 1 •If f(x) is an Nth degree polynomial then, dN + 1f ----------------= 0 N + 1 dx 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. N N – 1 fx a1x +++a2x aN + 1 p. 8.1 CE 30125 - Lecture 8 • 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 th • The approximation for the p derivative of some function fx can be found by th taking the p derivative of a polynomial approximation, gx, of the function fx. Procedure • Establish a polynomial approximation gx of degree N such that Np •gx is forced to be exactly equal to the functional value at N + 1 data points or nodes th th • The p derivative of the polynomial gx is an approximation to the p derivative of fx p. 8.2 CE 30125 - Lecture 8 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 f0 f1 f2 x x0 h x1 h x2 0 h h shifted x axis p. 8.3 CE 30125 - Lecture 8 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 2 gx= aox ++a1xa2 • The approximating Lagrange polynomial must match the functional values at all N + 1 data points or nodes (xo = 0 , x1 = h, x2 = 2h) 2 gxo = fo ao0 ++a10 a2 = fo 2 gx1 = f1 aoh ++a1ha2 = f1 2 gx2 = f2 ao2h ++a12h a2 = f2 p. 8.4 CE 30125 - Lecture 8 • Setting up the constraints as a system of simultaneous equations 0 0 1 ao fo 2 h h 1 a1 = f1 2 4h 2h 1 a2 f2 • Solve for ao , a1 , a2 f – 2f + f 4f –3f – f a = ---------------------------2 1 o- , a = ------------------------------1 2 o- , a = f o 2 1 2 o 2h 2h • The interpolating polynomial and its derivative are equal to: f2 – 2f1 + fo 2 4f1 – f2 – 3fo gx= ----------------------------- x ++-------------------------------- xfo 2h2 2h 1 f – 2f + f 4f – f – 3f gx = ----------------------------2 1 o- x + -------------------------------1 2 o- 2 h 2h p. 8.5 CE 30125 - Lecture 8 (1) Evaluating g ( x o ) to obtain a forward difference approximation to the first derivative • Evaluating the derivative of the interpolating function at xo = 0 1 1 g xo = g 0 1 –43fo + f1 – f2 g x = ------------------------------------ o 2h • Since the function fx is approximated by the interpolating function gx 1 1 f g xo xo 1 • Substituting in for the expression for g xo 1 –43fo + f1 – f2 f ------------------------------------ xo 2h p. 8.6 CE 30125 - Lecture 8 • Generalize the node numbering for the approximation x0 x1 x2 i i+1 i+2 generalized nodal numbering • This results in the generic 3 node forward difference approximation to the first deriva- tive at node i 1 – 3fi + 4fi + 1 – fi + 2 f ------------------------------------------------ i 2h p. 8.7 CE 30125 - Lecture 8 (1) Evaluating g ( x 1 ) to obtain a central difference approximation to the first derivative • Evaluating the derivative of the interpolating function at x1 = h 1 1 g x1 = g h 1 f2 – 2f1 + fo 4f1 – f2 – 3fo g x = ---------------------------------h + ------------------------------- 1 2 h 2h 1 f2 – fo g x = -------------- 1 2h • Again since the function fx is approximated by the interpolating function gx 1 1 f g x1 x1 1 • Substituting in for the expression for g x1 1 f2 – fo f -------------- x1 2h p. 8.8 CE 30125 - Lecture 8 • Generalize the node numbering x0 x1 x2 i-1 i i+1 • This results in the generic expression for the three node central difference approxima- tion to the first derivative 1 fi + 1 – fi – 1 f -------------------------- i 2h p. 8.9 CE 30125 - Lecture 8 (1) Evaluating g ( x 2 ) to obtain a backward difference approximation to the first derivative • Evaluating the derivative of the interpolating function at x2 = 2h 1 1 g x2 = g 2h 1 f – 2f + f 4f – f – 3f gx = --------------------------------2 1 o-2h + ------------------------------1 2 o- 2 2 h 2h 1 3f2 – 4f1 + fo g x = ------------------------------- 2 2h • Again since the function fx is approximated by the interpolating function gx 1 1 f g x2 x2 1 • Substituting in for the expression for g x2 1 3f2 – 4f1 + fo f ------------------------------- x2 2h p. 8.10 CE 30125 - Lecture 8 • Generalizing the node numbering x0 x1 x2 i-2 i-1 i • This results in the generic expression for a three node backward difference approxima- tion to the first derivative 1 3fi – 4fi – 1 + fi – 2 f ------------------------------------------- i 2h p. 8.11 CE 30125 - Lecture 8 (2) Evaluating g ( x o ) to obtain a forward difference approximation to the second derivative 2 • We note that in general g x can be computed as: f – 2f + f g2 x = ----------------------------2 1 o- 2 h • Evaluating the second derivative of the interpolating function at xo = 0 : 2 2 g xo = g 0 2 f2 – 2f1 + fo g xo = ---------------------------- h2 • Again since the function fx is approximated by the interpolating function gx, the second derivative at node xo is approximated as: 2 2 f g xo xo p. 8.12 CE 30125 - Lecture 8 2 • Substituting in for the expression for g xo f – 2f + f f2 ---------------------------2 1 o- x 2 o h • Generalizing the node numbering i i+1 i+2 x0 x1 x2 • This results in the generic expression for a three node forward difference approximation to the second derivative f – 2f + f f2 ---------------------------------------i + 2 i + 1 -i i 2 h p. 8.13 CE 30125 - Lecture 8 (2) Evaluating g ( x 1 ) to obtain a central difference approximation to the second derivative • Evaluating the second derivative of the interpolating function at x1 = h : 2 2 g x1 = g h 2 f2 – 2f1 + fo g x1 = ---------------------------- h2 • Again since the function fx is approximated by the interpolating function gx, the second derivative at node x1 is approximated as: 2 2 f g x1 x1 2 • Substituting in for the expression for g x1 f – 2f + f f2 ---------------------------2 1 o- x 1 h2 p. 8.14 CE 30125 - Lecture 8 • Generalizing node numbering i-1 i i+1 x0 x1 x2 • This results in the generic expression for a three node central difference approximation to the second derivative 2 f – 2f + f f ---------------------------------------i + 1 i i – 1- i 2 h Notes on developing differentiation formulae by interpolating polynomials • In general we can use any of the interpolation techniques to develop an interpolation function of degree Np . We can then simply differentiate the interpolating function and evaluate it at any of the nodal points used for interpolation in order to derive an approximation for the pth derivative. • Orders of accuracy may vary due to the accuracy of the interpolating function varying. • Exact accuracy can be obtained by substituting in Taylor series expansions or by consid- ering the accuracy of the approximating polynomial g(x). p. 8.15 CE 30125 - Lecture 8 Approximations and Associated Error Estimates to First and Second Derivatives Us- ing Quadratic Interpolation • We can derive an error estimate when using interpolating polynomials to establish finite difference formulae by simply differentiating the error estimate associated with the interpolating function. • We will illustrate the use of a 3 node Newton forward interpolation formula to derive: • A central approximation to the first derivative with its associated error estimate • A forward approximation to the second derivative with its associated error estimate Developing a 3 node interpolating function using Newton forward interpolation • A quadratic interpolating polynomial (N = 2 ) has 3 associated nodes (N +31 = ) or interpolating points. We again assume that the nodes are evenly distributed as: f0 f1 f2 x x0 x1 x2 hh • With a quadratic interpolating polynomial, we can derive differentiation formulae for both the first and second derivatives but no higher p. 8.16 CE 30125 - Lecture 8 • The approximating or interpolating function is defined using Newton forward interpola- tion as: fx= gx+ ex 2 f 1 f gx = f ++xx– -------o- ----- xx– xx– ----------o o o o 1 2 h 2! h xx– o xx– 1 xx– 2 3 ex= -------------------------------------------------------- f x x 3! o 2 • The error can be approximately expressed in either of the following forms: xx– o xx– 1 xx– 2 3 ex -------------------------------------------------------- f 3! o 3 xx– xx– xx– f ex --------------------------------------------------------o 1 2 ----------o 3 3! h • These latter two forms which do not involve are more suitable for the necessary differ- entiation w.r.t.