To Dream the Impossible Scheme Part 1 Approximating

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

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

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 How do we approximate f’(. 5)

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 2 -Point Forward Difference Approximation

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 2 -Point Backward Difference Approximation

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 2 -Point Central Difference Approximation

Approximating Derivatives from a Data Table x y=f(x) In Summary … so Far 0 2 0. 1 2. 204 Method 0. 2 2. 432 2 -PT BD 4. 44 0. 3 2. 708 2 -PT CD 5. 04 0. 4 3. 056 2 -PT FD 5. 64 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 Approximation Which is right? Which is better?

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 3 -PT FD Approx

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 4 -PT CD Approximation Note the new compact notation:

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 5 -PT FD Approximation:

Approximating Derivatives from a Data Table x y=f(x) In Summary 0 2 0. 1 2. 204 Method 0. 2 2. 432 2 -PT BD 4. 44 0. 3 2. 708 2 -PT CD 5. 04 0. 4 3. 056 2 -PT FD 5. 64 0. 5 3 -PT FD 4. 92 0. 6 4. 064 4 -PT CD 5. 00 0. 7 4. 772 5 -PT FD 5. 00 0. 8 5. 648 0. 9 6. 716 1 8 Approximation Which is the best approximation?

Approximating Derivatives from a Data Table x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 Method Approximation 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

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

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:

What’s Missing? Derivative Grid Scheme # Points 1 2 3 4 >=5 ☺ ☺ na na na 4 ? ? ? ☺ na na 5 ☺ ? ? ? ☺ >6 ? ? ? ? ? ? 2 ☺ na na 3 ? ? ? ☺ na na na 4 ☺ ? ? ? ☺ na na 5 ? ? ? ☺ na >6 ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 3 Forward/ Backward Difference Central Difference Non-Uniform Skewed-Grid Schemes

• 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:

Derivation of 2 -Point BD Equation for the 1 st Derivative on a Uniform Grid Start with Three 3 -Term Taylor Series Expansions. Where: fn=f(x 0+nδ) where δ is the grid spacing. Note: Equation for f 0 is expanded for use in further derivation Note: Define 00=1

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

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.

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

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

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

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

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 …

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

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

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

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.

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

Using Cramer's Rule to Solve for ωa 1

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

Turning our Attention to the Numerator … Minor of the Vandermonde Matrix With the (i+1)th row and nth column removed (from previous slide). Vandermonde Matrix with the rth row and nth column removed. Schur polynomial of order r-i-1 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.

Schur Polynomials

Therefore … Schur Polynomial det(V)

Finally … Where ωn is the nth weight for an r-point estimate of the ith derivative with grid points whose relative position to the center is given by {a 1, …, ar} and grid spacing is δ.

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.

Using Algorithm Generates … x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 3. 056 0. 5 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 1. 2 11. 312

It also Generates the 4 th Derivative… x y=f(x) 0 2 0. 1 2. 204 0. 2 2. 432 0. 3 2. 708 0. 4 0. 5 3. 056 3. 5 0. 6 4. 064 0. 7 4. 772 0. 8 5. 648 0. 9 6. 716 1 8 1. 2 11. 312

Derivative Grid Scheme # Points 2 3 Forward/ Backward Difference 4 5 >6 2 3 Central Difference 4 5 >6 Non-Uniform Skewed-Grid Schemes 1 2 3 4 >=5 ☺ ☺ ☺ na na ☺ ☺ na na na ☺ ☺ ☺ na na na ☺ ☺ ☺ ☺

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

Applying Finite Difference Schemes to Non-Rectangular Regions

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.

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

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

Analytic Solution Jm: Bessel Function of the First Kind of order m μmn: Is the nth eigenvalue of Jm

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.

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

Demonstration Using 3 -pt CD Formulations

Future Research Apply to More Complex Regions