To Dream the Impossible Scheme

Part 1Approximating Derivatives on Non-Uniform,

Skewed and Random Grid Schemes

Part 2Applying Rectangular Finite Difference Schemes to

Non-Rectangular Regions to Approximate

Solutions to Partial Differential Equations

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

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

Skewed

Non-Uniform

Random

Approximating Derivativesfrom 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 Derivativesfrom 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

'( ) 4.064 3.5(.5) 5.64

.1f

f x x f x

x

Approximating Derivativesfrom 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

( ) 3.5 3.056'(.5) 4.44.1

f x f x xf

x

Approximating Derivativesfrom 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

( ) 4.064 3.056'(.5) 5.042 .2

f x x f x xf

x

Approximating Derivativesfrom 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

In Summary … so Far

Method Approximation

2-PT BD 4.442-PT CD 5.042-PT FD 5.64

Which is right?

Which is better?

Approximating Derivativesfrom 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

3 4 ( ) ( 2 )'(.5) 4.92

2f x f x x f x x

fx

Approximating Derivativesfrom 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

2 1 1 28 8'(.5) 512

f f f ffx

Note the new compact notation:

( )nf f x n x

Approximating Derivativesfrom 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:

0 1 2 3 425 48 36 16 3'(.5) 512

f f f f ffx

Approximating Derivativesfrom 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

In Summary

Method Approximation

2-PT BD 4.442-PT CD 5.042-PT FD 5.643-PT FD 4.924-PT CD 5.005-PT FD 5.00

Which is the best approximation?

Approximating Derivativesfrom 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 4.442-PT CD 5.042-PT FD 5.643-PT FD 4.924-PT CD 5.005-PT CD 5.00

3

2

( ) 4 2 2

( ) 12 2 (.5) 5

f x x x

f x x f

Estimates of the 1st Derivative (CRC)

1 001( )

2hf x f f f

x

0 1 2

2(3)

01( ) 3 4

2 3hf x f f f f

x

2 1 1 2

4(5)

01( ) 8 8

12 30hf x f f f f f

x

0 1 2 3

4(5)

0 41( ) 25 48 36 16 3

512hf x f f f f f f

x

2-point FD:

3-point FD:

4-point CD:

5-point FD:

2-point CD: 1 1

2(3)

01( )

2 6hf x f f f

x

Estimates of Higher Order Derivatives (CRC)

2(4)

1 0 101( ) 2

12hf x f f f f

x

(3)3 2 130 0

1( ) 3 3f x f f f f O hx

44 3 2 1 040

1( ) 4 6 4 ( )f x f f f f f O hx

2nd D,2-point CD :

3rd D, 4-point FD:

3rd D, 4-point CD:

4th D, 5-point FD:

(3) 22 1 1 230

1( ) 2 22

f x f f f f O hx

4th D, 5-point CD: 4 2

2 1 0 1 2401( ) 4 6 4 ( )f x f f f f f O hx

What’s Missing?Derivative

Grid Scheme # Points 1 2 3 4 >=5

Forward/BackwardDifference

2 ☺ na na na na

3 ☺ ☺ na na na

4 ??? ??? ☺ na na

5 ☺ ??? ??? ☺>6 ??? ??? ??? ??? ???

CentralDifference

2 ☺ na na na na

3 ??? ☺ na na na

4 ☺ ??? ☺ na 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:

( )

0

( )( ) ( )!

nn

n

xf x x f xn

2 3 4 (4)( ) ( ) ( )( ) ( ) '( ) ( ) ( ) ( )

2! 3! 4!x x xf x x f x xf x f x f x f

1

(0) (1) (2) (3)0 0 0 0

2 3 4(4) ( )

2! 3! 4!f f f f f f

Derivation of 2-Point BD Equationfor the 1st Derivative on a Uniform Grid

Where: fn=f(x0+nδ) where δ is the grid spacing.Note: Equation for f0 is expanded for use in further derivationNote: Define 00=1

Start with Three 3-Term Taylor Series Expansions.

)3(0

3)2(

0

2)1(

0

1

0

0

0

)3(0

3)2(

0

2)1(

0

1

0

0

1

)3(0

3)2(

0

2)1(

0

1

0

0

2

!3

0

!2

0

!1

0

!0

0

!3!2!1!0

!3

2

!2

2

!1

2

!0

2

fffff

fffff

fffff

Derivation of 3-Point BD Equationfor the 1st Derivative on a Uniform Grid

Multiply Each Equation by a Weight ωn .

)2(0

2

0)1(

0

1

00

0

000

)2(0

2

1)1(

0

1

10

0

111

)2(0

2

2)1(

0

1

20

0

222

!2

0

!1

0

!0

0

!2!1!0

!2

2

!1

2

!0

2

ffff

ffff

ffff

Note: Error term dropped for the time being for brevity

Derivation of 3-Point BD Equationfor the 1st Derivative on a Uniform Grid

Sum up the Coefficients to Generate the 1st Derivative Expression .

0

!2

0

!2!2

2

1!1

0

!1!1

2

0!0

0

!0!0

2

2

0

2

1

2

2

1

0

1

1

1

2

0

0

0

1

0

2

Derivation of 3-Point BD Equationfor the 1st Derivative on a Uniform Grid

A little algebraic manipulation …

0012

!1012

0012

20

21

22

1

10

11

12

00

01

02

Derivation of 2-Point BD Equationfor the 1st Derivative on a Uniform Grid

Note: A Vandermonde Matrix

0

/!1

0

)0()1()2(

)0()1()2(

)0()1()2(1

0

1

2

222

111

000

And rewritten as a matrix equation …

Derivation of 3-Point BD Equationfor the 1st Derivative on a Uniform Grid

A General Vandermonde Matrix

113

12

11

223

22

21

321

11111

nn

nnn

n

n

V

nji

ijV1

)()det(

Solving for ω-2 Using Cramer’s Rule

0 0

1 1 0 03

2 2 2 2

2 0 0 0

1 1 1

2 2 2

0 1 0

1! 1 0 1 01! 1

0 1 0 1 0 1

( 1 2)(0 2)(0 1) 22 1 0

2 1 0

2 1 0

Cofactor Expansion

Determinant of a Vandermonde matrix

Derivation of 3-Point BD Equationfor the 1st Derivative on a Uniform Grid

Solve for the Remaining Weights.

2

3,

2

4,

2

1012

3

3 23 (3) (3)

2 1 03

( 2)

[ ] ( 1) ( ) ( )3! 3

0

R f f

Now use weights to calculate the coefficient of the remainder term …

Derivation of 3-Point BD Equationfor the 1st Derivative on a Uniform Grid

Voila! .

2

(1) (3)0 2 1 0

14 3 ( )

2 3f f f f f

Derivation of 3-Point BD Equationfor the 2nd Derivative on a Uniform Grid

Alter RHS Slightly ….

2

0 0 0

21 1 1

1

2 2 2 0

2 1 0

2 1 0

2 1 1

0 01! 00 2!

Derivation of 5-Point CD Equationfor the 3rd Derivative on a Uniform GriD

(or, if I desire, anything up to the 4th Derivative)

3

4

0 0

1 1

0 0 0

1 1 1 2

12 2 2 2 2

3 3 3 3 3 1

24 4 4 4 4

0

1 2

01 2001 2

3!/ 02 1 0

2 1 0

02 1 0002 1 0

1 2 0 4!/

2 1 0 1 2

(or, if I desire, anything up to the 4th Derivative)

System will also Work for Skew Grid Schemes(i.e. use backward 1st and 4th point and forward 1st , 2nd, and 6th point

to find the 3rd derivative on a uniform grid)

3

0 0 0 0 0

1 1 1 1 1 4

12 2 2 2 2

1

3 3 3 3 3 2

64 4 4 44

2

4 1 1 2 6

04 1 1 2 6004 1 1 2 6

3!/4 1 1 2 6 0

4 1 1 6

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).

1

0 0 0

1

1

1 1 1

1

0

!/

0

i

r

ai r

i i i iai r

r r rai r

a a a

ia a a

a a a

Using Cramer's Rule to Solve for ωa1

1

0 0

1 1

0 0 0

1

1

1 1 1

1

0

!/

0

i r

i iii r

r r

i r

a

i r

i i i

i r

r r r

i r

a a

i a a

a a

a a a

a a a

a a a

Which “Simplifies” to:

1

0 0 0

2

1 1 1 1

2

1 1 1

2

1

!1

i r

i n i i i

i i r

r r r

i r

a

j kk j r

a a a

ia a a

a a a

a a

Determinant of a Vandermonde matrixCofactor ExpansionAbout the 1st 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) 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 computingtheir coefficients, The Mathematical and Computer Modeling, 34 (2003) 280-287.

1 11,det , ..., , ..., detr i m n ri nM s a a a V

Minor of the Vandermonde MatrixWith the (i+1)th row and nth columnremoved (from previous slide).

Schur polynomialof order r-i-1

Vandermonde Matrix withthe rth row and nth columnremoved.

Schur Polynomials

0 1

1 11

2 11

3 11

11

,..., 1

,...,

,...,

,...,

,...,

n

n

n jj

n j kj k n

n j k lj k ln

n

n n jj

s a a

s a a a

s a a a a

s a a a a a

s a a a

Therefore …

111

1 ,

1

,..., ,...,

!1

m n r j kr ik j r

i n j k nn i

j kk j r

s a a a a a

i

a a

det V

det(V)

Schur Polynomial

Finally …

11 1

1

,..., ,...,!1

m n ri n r i

n in j

j n r

s a a ai

a a

Where ωn is the nth weight for an r-point estimate of theith derivative with grid points whose relative position tothe center is given by {a1, …, ar} and grid spacing is δ.

Recall the Earlier Example …(i.e. use backward 1st and 4th point and forward 1st , 2nd, and 6th point

to estimate the 3rd derivative on a uniform grid)

3

0 0 0 0 0

1 1 1 1 1 4

12 2 2 2 2

1

3 3 3 3 3 2

64 4 4 44

2

4 1 1 2 6

04 1 1 2 6004 1 1 2 6

3!/4 1 1 2 6 0

4 1 1 6

Note: The grid is “uniform”, the spacing between the points is not.

Using Algorithm Generates …

5(3) 24 1 1 2 63

1 4 5 9 1 3 21

75 21 25 6 350 20f f f f f f f

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

(3) 1(.5) ( .11755 .727619 1.46304 .795333 .081634) 24.001

f

3

2

( ) 24 24

( ) 4 2 2

( ) 12 2

x x f x

f x x x

f x x

f

It also Generates the 4th Derivative…

5(4) 24 1 1 2 64

1 2 4 12 1 3 4

75 21 25 3 175 5f f f f f f f

x y=f(x)

0 2

0.1 2.204

0.2 2.432

0.3 2.708

0.4 3.0560.5 3.5

0.6 4.064

0.7 4.772

0.8 5.648

0.9 6.716

1 8

1.2 11.312

(3) 1(.5) (.058773 .5821 1.95072 1.59067 .163269) 3.61 13.0001

f E

(4)

3

2

( ) 0( ) 24 24

( ) 4 2 2

( ) 12 2

xx x f x f

f x x x

f x x

f

Derivative

Grid Scheme # Points 1 2 3 4 >=5

Forward/BackwardDifference

2 ☺ na na na na

3 ☺ ☺ na na na

4 ☺ ☺ ☺ na na

5 ☺ ☺ ☺ ☺>6 ☺ ☺ ☺ ☺ ☺

CentralDifference

2 ☺ na na na na

3 ☺ ☺ na na na

4 ☺ ☺ ☺ na na

5 ☺ ☺ ☺ ☺>6 ☺ ☺ ☺ ☺ ☺

Non-Uniform ☺ ☺ ☺ ☺ ☺Skewed-Grid Schemes ☺ ☺ ☺ ☺ ☺

The Extension to Random Grids…

11 1

1

,..., ,...,!1

m n ri n r i

n in j

j n r

s a a ai

a a

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

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5

-4 -3 -2 -1 0 1 2 3 4

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

The Wave Equationon a Circular Membrane

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

Initial Condition

Object: Solve analytically using the polar from of the wave equation.Then compare to a numerical finite difference approximation thatsuperimposes a rectangular grid on the circle. Note that the grid size varies from point to point on the circle.

The Wave Equation

udt

u 222

2

2

2

2

22

2

2

dy

u

dx

u

dt

u

2

2

22

2

2 111

d

u

rr

u

rrrdt

u

2

22

2

2 1

dr

u

r

u

rdt

u

Rectangular Form:

Wave Equation:

Polar Form:

Polar Form: (Radial Symmetry)

Boundary/Initial Conditions

continuity),2,('),0,('

continuity),2,(),0,(

continuity),,0(

membranepinned0),,1(

trutru

trutru

tu

tu

velocityinitialno0)0,,('

)sin(8216)0,,(

ru

rrru

2

2

211

2

2

2

2

d

u

rr

u

rdr

u

dt

u

PDE (ω=1, 0≤r ≤1):

Boundary Conditions:

Initial Conditions:

Analytic Solution

20

10

)(2

cos

)(2

sin2)(

20

10 )cos(

)sin()()sin(8

216

rdrdm

mrmnmJ

rdrdm

mrmnmJrr

mnB

mnA

Jm: Bessel Function of the First Kind of order mμmn: Is the nth eigenvalue of Jm

)cos()sin(

)cos(0 1

mm

tmnrmnmJm n mnB

mnAu

Numeric Solution

kjiy

kjiy

kjiy

kjix

kjix

kjix

kji

kji

kji

uuuuuu

uuut

1,,1,,,,1

1,,

1,2

101101

21

Since the grid is rectangular, use the rectangular form of thewave equation:

2

2

2

2

2

2

dy

u

dx

u

dt

u

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

kjiy

kjiy

kjiy

kjix

kjix

kjix

kji

kji

kji

uuuuuut

uuu

1,,1,,,1,1

2

,,1

,

101101

2

Time Stepping:

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

Stability Requirement:Δt ≤ smallest grid increment

DemonstrationUsing 3-pt CD Formulations

Future ResearchApply to More Complex Regions

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5

-4 -3 -2 -1 0 1 2 3 4

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3