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 Skewed Non-Uniform Random

Approximating Derivatives from a Data Table xy=f(x) How do we approximate f’(.5)

Approximating Derivatives from a Data Table xy=f(x) Point Forward Difference Approximation

Approximating Derivatives from a Data Table xy=f(x) Point Backward Difference Approximation

Approximating Derivatives from a Data Table xy=f(x) Point Central Difference Approximation

Approximating Derivatives from a Data Table xy=f(x) In Summary … so Far MethodApproximation 2-PT BD PT CD PT FD 5.64 Which is right? Which is better?

Approximating Derivatives from a Data Table xy=f(x) PT FD Approx

Approximating Derivatives from a Data Table xy=f(x) PT CD Approximation Note the new compact notation:

Approximating Derivatives from a Data Table xy=f(x) PT FD Approximation:

Approximating Derivatives from a Data Table xy=f(x) In Summary MethodApproximation 2-PT BD PT CD PT FD PT FD PT CD PT FD 5.00 Which is the best approximation?

Approximating Derivatives from a Data Table xy=f(x) MethodApproximation 2-PT BD PT CD PT FD PT FD PT CD PT CD 5.00

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

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# 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 ???

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 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.

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

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 Note: A Vandermonde Matrix And rewritten as a matrix equation …

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 GriD (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 i th derivative) a n : integer that describes position of grid point with respect to center point (i.e. a n Δx).

Using Cramer's Rule to Solve for ω a 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 … T. Ernst, Generalized Vendermonde Systems of Equations. Mathematics of Computation, 24, (1970) I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Mongraphs, Second Ed S.D. Marchi, Polynomials arising in factoring generalized Vandermonde determinants: An algorthm for computing their coefficients, The Mathematical and Computer Modeling, 34 (2003) 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.

Schur Polynomials

Therefore … det(V) Schur Polynomial

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.

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 … xy=f(x)

It also Generates the 4 th Derivative… xy=f(x)

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 ☺☺☺☺☺

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.

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: Wave Equation: Polar Form: (Radial Symmetry)

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

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

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