1. To Dream the Impossible Scheme Part 1 Approximating Derivatives on Non-Uniform, Skewed and Random Grid Schemes Part 2 Applying Rectangular Finite Difference Schemes to Non-Rectangular Regions to Approximate Solutions to Partial Differential Equations

2. Approximating Derivatives on Non-Uniform, Skewed and Random, Grid Schemes Skewed Non-Uniform Random

3. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 How do we approximate f’(.5)

4. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 2-Point Forward Difference Approximation

5. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 2-Point Backward Difference Approximation

6. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 2-Point Central Difference Approximation

7. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 In Summary … so Far MethodApproximation 2-PT BD 4.44 2-PT CD 5.04 2-PT FD 5.64 Which is right? Which is better?

8. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 3-PT FD Approx

9. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 4-PT CD Approximation Note the new compact notation:

10. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 5-PT FD Approximation:

11. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 In Summary MethodApproximation 2-PT BD 4.44 2-PT CD 5.04 2-PT FD 5.64 3-PT FD 4.92 4-PT CD 5.00 5-PT FD 5.00 Which is the best approximation?

12. Approximating Derivatives from a Data Table xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 MethodApproximation 2-PT BD 4.44 2-PT CD 5.04 2-PT FD 5.64 3-PT FD 4.92 4-PT CD 5.00 5-PT CD 5.00

13. Estimates of the 1 st Derivative (CRC) 2-point FD: 3-point FD: 4-point CD: 5-point FD: 2-point CD:

14. Estimates of Higher Order Derivatives (CRC) 2 nd D,2-point CD : 3 rd D, 4-point FD: 3 rd D, 4-point CD: 4 th D, 5-point FD: 4 th D, 5-point CD:

15. What’s Missing? Derivative Grid Scheme# Points1234>=5 Forward/ Backward Difference 2 ☺ na 3 ☺☺ 4 ??? ☺ na 5 ☺ ??? ☺ >6 ??? Central Difference 2 ☺ na 3 ??? ☺ na 4 ☺ ??? ☺ na 5 ??? ☺ na >6 ??? Non-Uniform ??? Skewed-Grid Schemes ???

16. Where do these Equations Come From –Derivation starts with the Taylor Series centered on x: –i.e: –Or in a shorthand form the you will see on the following slides:

17. Derivation of 2-Point BD Equation for the 1 st Derivative on a Uniform Grid Where: f n =f(x 0 +nδ) where δ is the grid spacing. Note: Equation for f 0 is expanded for use in further derivation Note: Define 0 0 =1 Start with Three 3-Term Taylor Series Expansions.

18. Derivation of 3-Point BD Equation for the 1 st Derivative on a Uniform Grid Multiply Each Equation by a Weight ω n. Note: Error term dropped for the time being for brevity

19. Derivation of 3-Point BD Equation for the 1 st Derivative on a Uniform Grid Sum up the Coefficients to Generate the 1 st Derivative Expression.

20. Derivation of 3-Point BD Equation for the 1 st Derivative on a Uniform Grid A little algebraic manipulation …

21. Derivation of 2-Point BD Equation for the 1 st Derivative on a Uniform Grid Note: A Vandermonde Matrix And rewritten as a matrix equation …

22. Derivation of 3-Point BD Equation for the 1 st Derivative on a Uniform Grid A General Vandermonde Matrix

23. Solving for ω -2 Using Cramer’s Rule Cofactor Expansion Determinant of a Vandermonde matrix

24. Derivation of 3-Point BD Equation for the 1 st Derivative on a Uniform Grid Solve for the Remaining Weights. Now use weights to calculate the coefficient of the remainder term …

25. Derivation of 3-Point BD Equation for the 1 st Derivative on a Uniform Grid Voila !.

26. Derivation of 3-Point BD Equation for the 2 nd Derivative on a Uniform Grid Alter RHS Slightly ….

27. Derivation of 5-Point CD Equation for the 3 rd Derivative on a Uniform GriD (or, if I desire, anything up to the 4 th Derivative)

28. System will also Work for Skew Grid Schemes (i.e. use backward 1 st and 4 th point and forward 1 st, 2 nd, and 6 th point to find the 3 rd derivative on a uniform grid) Note: The grid is “uniform”, the spacing between the points is not.

29. A General Matrix System (for an r-point approximation for the i th derivative) a n : integer that describes position of grid point with respect to center point (i.e. a n Δx).

30. Using Cramer's Rule to Solve for ω a 1

31. Which “Simplifies” to: Determinant of a Vandermonde matrix Cofactor Expansion About the 1 st Column and The (i+1) th Row

32. Turning our Attention to the Numerator … T. Ernst, Generalized Vendermonde Systems of Equations. Mathematics of Computation, 24, (1970) 893-903. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Mongraphs, Second Ed. 1995. S.D. Marchi, Polynomials arising in factoring generalized Vandermonde determinants: An algorthm for computing their coefficients, The Mathematical and Computer Modeling, 34 (2003) 280-287. Minor of the Vandermonde Matrix With the (i+1) th row and n th column removed (from previous slide). Schur polynomial of order r-i-1 Vandermonde Matrix with the r th row and n th column removed.

33. Schur Polynomials

34. Therefore … det(V) Schur Polynomial

35. Finally … Where ω n is the n th weight for an r-point estimate of the i th derivative with grid points whose relative position to the center is given by {a 1, …, a r } and grid spacing is δ.

36. Recall the Earlier Example … (i.e. use backward 1 st and 4 th point and forward 1 st, 2 nd, and 6 th point to estimate the 3 rd derivative on a uniform grid) Note: The grid is “uniform”, the spacing between the points is not.

37. Using Algorithm Generates … xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 1.211.312

38. It also Generates the 4 th Derivative… xy=f(x) 02 0.12.204 0.22.432 0.32.708 0.43.056 0.53.5 0.64.064 0.74.772 0.85.648 0.96.716 18 1.211.312

39. Derivative Grid Scheme# Points123 4>=5 Forward/ Backward Difference 2 ☺ na 3 ☺☺ 4 ☺☺☺ 5 ☺☺☺☺ >6 ☺☺☺☺☺ Central Difference 2 ☺ na 3 ☺☺ 4 ☺☺☺ 5 ☺☺☺☺ >6 ☺☺☺☺☺ Non-Uniform ☺☺☺☺☺ Skewed-Grid Schemes ☺☺☺☺☺

40. The Extension to Random Grids… A slight adjustment to this equation will accomplish this. Let δ=1 and a i be the position from the point of interest.

41. Applying Finite Difference Schemes to Non-Rectangular Regions

42. The Wave Equation on a Circular Membrane Object: Solve analytically using the polar from of the wave equation. Then compare to a numerical finite difference approximation that superimposes a rectangular grid on the circle. Note that the grid size varies from point to point on the circle.

43. The Wave Equation Rectangular Form: Wave Equation: Polar Form: (Radial Symmetry)

44. Boundary/Initial Conditions PDE ( ω=1, 0≤r ≤1 ): Boundary Conditions: Initial Conditions:

45. Analytic Solution J m : Bessel Function of the First Kind of order m μ mn : Is the n th eigenvalue of J m

46. Numeric Solution Since the grid is rectangular, use the rectangular form of the wave equation: The discrete form of this equation from finite difference methods Note: Based on 3-point central difference formulations of the spatial terms. Note: Based on 3-point backward difference formulation in time. Note: The time grid is uniform.

47. Numeric Solution Time Stepping: Stability Requirement: Δt ≤ smallest grid increment

48. Demonstration Using 3-pt CD Formulations

50. Future Research Apply to More Complex Regions