o introduction to walsh functions in numerical...

Walsh Functions inNumericalSimulation

Peter A. Gno↵o

Introduction toWalsh functions

Preliminaries andperspectives

Fractal derivation

Walsh Functiondefinition

Series expansionsof functions

Series expansion offunction reciprocal

Integrals of seriesexpansions

Derivatives ofseries expansions

Series expansionsof functions of twovariables

Partial di↵erentialequation solutions

Burger’s Equation

Advection run times

Riemann Problem -Sod

Future work

Walsh Functions in Numerical SimulationA New Framework for Solving Nonlinear Systems of

PDEs

Peter A. Gno↵o

NASA Langley Research CenterHampton, Virginia

April 22, 2014


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

First eight Walsh functions

x0 0.25 0.5 0.75 1

-1

0

1

g1(x)

x0 0.25 0.5 0.75 1

-1

0

1

g5(x)

x0 0.25 0.5 0.75 1

-1

0

1

g2(x)

x0 0.25 0.5 0.75 1

-1

0

1

g6(x)

x0 0.25 0.5 0.75 1

-1

0

1

g3(x)

x0 0.25 0.5 0.75 1

-1

0

1

g7(x)

x0 0.25 0.5 0.75 1

-1

0

1

g4(x)

x0 0.25 0.5 0.75 1

-1

0

1

g8(x)


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Walsh Function Philosophical Perspectives

I Particle - wave duality in nature is intrinsic to a Walshseries representation of a function.

I Fractal patterns in nature (infinitely self-similar) arereflected in the fractal derivation of Walsh functions.

I “Continuity” is a powerful mathematical construct butan illusion in nature. Taylor series require continuity.Walsh series do not assume continuity.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Walsh Function Practical Perspectives

I Closure of non-linear problems is enabled by theself-mapping property under multiplication.

I Global support (interior and boundary) of every solutioncomponent imparts complexity and simplicity to thesolution of PDEs.

I Walsh functions utilize a grid-free approach in anexplicit sense though segment lengths implicitly define adiscretization of space and time.

I A “derivative” is a global construct in Walsh functionsas opposed to a local construct in Taylor series.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Preliminaries

I Let g

n

(x) be a basis function over the domainx

a

x x

b

with n piecewise constant segments. Theorthonormal basis function requires:

Zx

b

x

a

g

n

(x)gm

(x)dx = �nm

I A constraint of equal segment lengths leads to analgorithm that rapidly becomes intractable for large n;fractional overlap of segments complicates the algebra.

I A more productive approach is to let index p � 0 definea segment size dx

p

.

dx

p

= (xb

� x

a

)/2p

I Apply a new constraint on segment lengths such that atmost two segment lengths (dx

p

and 2dx

p

) are allowedto create n segments spanning the domain.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Fractal Derivation: Segment Counts

Let n1 equal the number of segments with length dx

p

and n2

equal the number of segments with length 2dx

p

in thedistribution vector g

n

. Then

n1 + n2 = n

n1 + 2n2 = 2p

so that

n1 = 2n � 2p

n2 = 2p � n


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Fractal Derivation: Origami Analogy

I Starting with flat strip of paper, sequentially fold it inhalf p times.

I The first fold produces 1 crease at the center of thestrip (x = 1/2); the second fold produces 2 creases atx=1/4 and 3/4; the third fold produces 4 creases at x= 1/8, 3/8, 5/8, and 7/8; ...

I Assign segments in order of increasing crease number.If the number of remaining segments is odd then placea segment of length 2dx

p

across the crease. If thenumber of remaining segments is even then place twosegments of length dx

p

on both sides of the crease.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Segment Size Distribution for First 32 Functionsn p

1................................ 1 01 1............................... 2 1

1 2 1.............................. 3 21 1 1 1............................. 4 2

1 2 2 2 1............................ 5 31 2 1 1 2 1........................... 6 3

1 1 1 2 1 1 1.......................... 7 31 1 1 1 1 1 1 1......................... 8 3

1 2 2 2 2 2 2 2 1........................ 9 41 2 2 2 1 1 2 2 2 1....................... 10 4

1 2 1 1 2 2 2 1 1 2 1...................... 11 41 2 1 1 2 1 1 2 1 1 2 1..................... 12 4

1 1 1 2 1 1 2 1 1 2 1 1 1.................... 13 41 1 1 2 1 1 1 1 1 1 2 1 1 1................... 14 4

1 1 1 1 1 1 1 2 1 1 1 1 1 1 1.................. 15 41 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1................. 16 4

1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1................ 17 51 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 1............... 18 5

1 2 2 2 1 1 2 2 2 2 2 2 2 1 1 2 2 2 1.............. 19 51 2 2 2 1 1 2 2 2 1 1 2 2 2 1 1 2 2 2 1............. 20 5

1 2 1 1 2 2 2 1 1 2 2 2 1 1 2 2 2 1 1 2 1............ 21 51 2 1 1 2 2 2 1 1 2 1 1 2 1 1 2 2 2 1 1 2 1........... 22 5

1 2 1 1 2 1 1 2 1 1 2 2 2 1 1 2 1 1 2 1 1 2 1.......... 23 51 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1......... 24 5

1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 1........ 25 51 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1....... 26 5

1 1 1 2 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 2 1 1 1...... 27 51 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1..... 28 5

1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 1.... 29 51 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1... 30 5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.. 31 51 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 32 5


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Segment Layout and Fractal Patterns

bi-section

n

4 4

8 8

12 12

16 16

20 20

24 24

28 28

32 32

36 36

40 40

44 44

48 48

52 52

56 56

60 60

64 64

12 20 03 3 3 3


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Color Coded Segments by Sign


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Walsh Function Definition

g

n

(x) =

8>>>><

>>>>:

(�1)(m+1)(xb

� x

a

)�1/2 for x

m�1 < x < x

m

0 for x = x

m

(xb

� x

a

)�1/2 for x = x

a

(�1)(n+1)(xb

� x

a

)�1/2 for x = x

b

where

x

m

= x

a

+mX

i=1

g

n

(i)dx

p

and i , m, and n are positive integers.

g

n

(i) =

(1 if x

i

� x

i�1 = dx

p

2 if x

i

� x

i�1 = 2dx

p

where 1 i n.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Recursion Relations on the Half Domain

g

n

(x) =hg

L

k

(x), (�1)(k+n)g

R

k

(x)i/p

2

k = (n + modulo(n, 2))/2

g1 = [gL

1 , gR

1 ]/p

2

g2 = [gL

1 ,�gR

1 ]/p

2

g3 = [gL

2 ,�gR

2 ]/p

2

g4 = [gL

2 , gR

2 ]/p

2

g5 = [gL

3 , gR

3 ]/p

2

g6 = [gL

3 ,�gR

3 ]/p

2

g7 = [g l

4,�gR

4 ]/p

2

g8 = [gL

4 , gR

4 ]/p

2

g9 = [gL

5 , gR

5 ]/p

2

g10 = [gL

5 ,�gR

5 ]/p

2

g11 = [gL

6 ,�gR

6 ]/p

2

g12 = [gL

6 , gR

6 ]/p

2

g13 = [gL

7 , gR

7 ]/p

2

g14 = [gL

7 ,�gR

7 ]/p

2

g15 = [gL

8 ,�gR

8 ]/p

2

g16 = [gL

8 , gR

8 ]/p

2


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Multiplication Table P(n, m) - Closure

1p(x

b

�x

a

)g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16

g1 g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 g12 g13 g14 g15 g16

g2 g1 g4 g3 g6 g5 g8 g7 g10 g9 g12 g11 g14 g13 g16 g15

g3 g1 g2 g7 g8 g5 g6 g11 g12 g9 g10 g15 g16 g13 g14

g4 g1 g8 g7 g6 g5 g12 g11 g10 g9 g16 g15 g14 g13

g5 g1 g2 g3 g4 g13 g14 g15 g16 g9 g10 g11 g12

g6 g1 g4 g3 g14 g13 g16 g15 g10 g9 g12 g11

g7 g1 g2 g15 g16 g13 g14 g11 g12 g9 g10

g8 g1 g16 g15 g14 g13 g12 g11 g10 g9

g9 g1 g2 g3 g4 g5 g6 g7 g8

g10 g1 g4 g3 g6 g5 g8 g7

g11 g1 g2 g7 g8 g5 g6

g12 g1 g8 g7 g6 g5

g13 g1 g2 g3 g4

g14 g1 g4 g3

g15 g1 g2

g16 g1


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Series Expansion of a FunctionA function f (x) may be represented by an infinite series

1X

n=1

A

n

g

n

(x)

over the interval x

a

x x

b

. The coe�cients A

n

arecomputed

A

n

=

Zx

b

x

a

f (x)gn

(x)dx

Because g

n

(x) is a constant on any given segment andchanges sign across segments, the integral may be expressedas

A

n

= (xb

� x

a

)�1/2nX

i=1

(�1)(i+1)

Zx

i+1

x

i

f (x)dx

(Eventually replaced by a Fast Walsh Transform!)


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: f (x) = x

x

f(x)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

An gn(x)f(x) = x

n

An

8 16 24 32 40 48 56 64-1

-0.8

-0.6

-0.4

-0.2

0


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: f (x) = 12x

2

x

f(x)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

An gn(x)f(x) = 0.5 x2

n

An

8 16 24 32 40 48 56 640

0.05

0.1

0.15

0.2

0.25


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: f (x) = cos(x)

x

f(x)

-2 0 2-1

-0.5

0

0.5

1

An gn(x)f(x) = cos(x)

n

An

8 16 24 32 40 48 56 64

-1.5

-1

-0.5

0


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: f (x) = 1� H(x � 13)

x

f(x)

-1 -0.5 0 0.5 1-0.5

0

0.5

1

1.5

An gn(x)f(x) = 1 - H(x-1/3)

n

An

8 16 24 32 40 48 56 64-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(No Gibbs phenomenon!)


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Series Expansion of Function ReciprocalIf a function f (x) is represented by the truncated series

2pX

n=1

A

n

g

n

(x)

and its reciprocal, r(x) = 1/f (x), is represented by the series

2pX

m=1

B

m

g

m

(x)

then the coe�cients B

m

can be computed as a function ofcoe�cients A

n

.

B

m

=⇥AP(n,m)

⇤�1�(n, 1)(x

b

� x

a

)

(Execute with Fast Walsh Reciprocal Transform!)


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: r(x) = 1/x from f (x) = x

x

r(x)

-0.1 -0.05 0 0.05 0.1 0.15 0.2-100

-50

0

50

100

Bm gm(x)r(x) = 1/x

p = 6, n = 64

x

r(x)

-0.1 -0.05 0 0.05 0.1 0.15 0.2-100

-50

0

50

100

Bm gm(x)r(x) = 1/x

p = 10, n = 1024

A value of 0 in the domain is not a problem!


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: r(x) = 1/x2 from f (x) = x

2

x

r(x)

-0.1 0 0.1 0.20

200

400

600

800

1000Bm gm(x)r(x) = 1/x2

p = 6, n = 64

x

r(x)

-0.1 -0.05 0 0.05 0.1 0.15 0.20

200

400

600

800

1000Bm gm(x)r(x) = 1/x2

p = 10, n = 1024


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Integrals of Series Expansions

Let@f

@x

(x) = f

x

(x) =1X

n=1

A

n

g

n

(x).

It follows that

f (x) = f (xa

)+

Zx

x

a

1X

n=1

A

n

g

n

(s)ds = f (xa

)+1X

n=1

A

n

Zx

x

a

g

n

(s)ds.

Define an integrating matrix �(n,m) such that

Zx

x

a

g

n

(s)ds = (xb

� x

a

)1X

m=1

�(n,m)gm

(x)

This integral looks like a saw with the pointed teeth locatedat every segment boundary.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Integrating Matrix �(n, m)

�(n,m) =

2

66666666666664

12

�14 0 �1

8 0 0 0 �116 · · ·

14 0 �1

8 0 0 0 �116 0 · · ·

0 18 0 0 0 �1

16 0 0 · · ·18 0 0 0 �1

16 0 0 0 · · ·0 0 0 1

16 0 0 0 0 · · ·0 0 1

16 0 0 0 0 0 · · ·0 1

16 0 0 0 0 0 0 · · ·116 0 0 0 0 0 0 0 · · ·...

......

......

......

.... . .

3

77777777777775


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Coe�cients of Integral from Integrand

f (x) = f (xa

)+(xb

�x

a

)1X

m=1

1X

n=1

A

n

�(n,m)gm

(x) =1X

m=1

A

m

g

m

(x)

where

A

m

= (xb

� x

a

)�T (m, n)An

+ �1m

(xb

� x

a

)1/2f (x

a

)


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: f (x) = cos(x) from f

x

(x) = � sin(x)

n

An

8 16 24 32 40 48 56 64-2

-1

0

1

2

3

p = 6p = 6 integrating matrixp = 7 integrating matrixp = 8 integrating matrix

Series coe�cients computeddirectly and indirectly with

the integrating matrix.

n|∆

An|/

|An

8 16 24 32 40 48 56 640.0000

0.0005

0.0010

0.0015

p = 6p = 7p = 8

Di↵erence between seriescoe�cients computed directly

and indirectly.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Derivatives of Series Expansions

A

m

= (xb

� x

a

)�T (m, n)An

+ �1m

(xb

� x

a

)1/2f (x

a

)

The truncated series coe�cients for f

x

(x) are now derivedfrom the truncated series coe�cients of f (x).

A

n

= (xb

� x

a

)�1[�T (m, n)]�1hA

m

� �1m

(xb

� x

a

)1/2f (x

a

)i

A

n

= (xb

� x

a

)�1Dp(n,m)hA

m

� �1m

(xb

� x

a

)1/2f (x

a

)i

The derivative has global support. A local, divided di↵erencehas no meaning in the context of a Walsh function

representation.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

The Di↵erentiating Matrix Dp(n, m)

D2 =

2

664

0 0 0 �80 0 �8 00 8 0 �168 0 16 32

3

775

D3 =

2

66666666664

0 0 0 0 0 0 0 �160 0 0 0 0 0 �16 00 0 0 0 0 �16 0 00 0 0 0 �16 0 0 00 0 0 16 0 0 0 �320 0 16 0 0 0 �32 00 16 0 0 0 32 0 �64

16 0 0 0 32 0 64 128

3

77777777775


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

The Role of the Integration Constant

Di↵erentiate the series representation of

f (x) = sin(x) =2pX

m

A

m

g

m

(x)

with C = f (xa

) to compute the series coe�cients for

f

x

(x) = cos(x) =2pX

n

A

n

g

n

(x).

Set C = 0.

x

f(x

)

-3 -2 -1 0 1 2 3

-2

-1

0

1

2


~

C = 0

Choose C such that

˜A2p

= 0.

x

f(x

)

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

-0.5

0

0.5

1


A32(C) = 0~

~


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Series Expansions of Functions of Two Variables

A function f (x , y) may be represented by an infinite series

1X

n=1

1X

m=1

A

mn

g

m

(y)gn

(x)

over the interval x

a

x x

b

and y

a

y y

b

. Thecoe�cients A

mn

are computed

A

mn

=

Zx

b

x

a

Zy

b

y

a

f (x , y)gm

(y)gn

(x)dxdy

Use Fast Walsh transform in two dimensions.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Example: f (x , y) = (x � y)3 + (x + y)2

x

-1

-0.5

0

0.5

1

y

-1-0.5

00.5

1

f

-5

0

5


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Burger’s Equation: u

t

+ 12(u

2)x

= ⌫u

xx

The basis function series representation of u(x , t) is

u(x , t) =N↵X

j=1

N⌧X

i=1

U(i , j)gi

(t)gj

(x).

The basis function series for @u(x ,t)@x

= u

x

(x , t) is

u

x

(x , t) =N↵X

j=1

N⌧X

i=1

U

x

(i , j)gi

(t)gj

(x).

The coe�cients U

x

(i , j) may be expressed as a function ofthe coe�cients U(i , j).

U

x

(i , j) = L�1x

[Dp↵(j , ↵)U(i , ↵)�D

p↵(j , 1)Ux0(i)]


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Burgers Equation(2)

The basis function series for @u(x ,t)@t

= u

t

(x , t) is

u

t

(x , t) =N↵X

j=1

N⌧X

i=1

U

t

(i , j)gi

(t)gj

(x).

The coe�cients U

t

(i , j) are expressed as a function of thecoe�cients U(i , j) and constant of integration U

t0(j).

U

t

(i , j) = L�1t

[Dp⌧ (i , ⌧)U(⌧, j)�D

p⌧ (i , 1)Ut0(j)]


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Burgers Equation(3)

The non-linear component u

2(x , t) is

u

2(x , t) =

N⌧X

⌧1=1

N↵X

↵1=1

U(⌧1, ↵1)g⌧1(t)g↵1(x)

!⇥

N⌧X

⌧2=1

N↵X

↵2=1

U(⌧2, ↵2)g⌧2(t)g↵2(x)

!

u

2(x , t) =N⌧X

⌧1=1

N↵X

↵1=1

N⌧X

⌧2=1

N↵X

↵2=1

U(⌧1, ↵1)U(⌧2, ↵2)⇥

(g⌧1(t)g⌧2(t))(g↵1(x)g↵2(x))

u

2(x , t) = (Lt

Lx

)�1/2U(⌧1, ↵1)U(⌧2, ↵2)⇥

(�(i ,P(⌧1, ⌧2))gi

(t))(�(j ,P(↵1, ↵2))gj

(x))


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Burgers Equation(4)Each component factor of g

i

(t)gj

(x) must equal zerobecause the functions are mutually orthogonal.Consequently, there are N⌧N↵ residual equations defined by

r(i , j) = U

t

(i , j) + F

x

(i , j)

= [Dp⌧ (i , ⌧)U(⌧, j)�D

p⌧ (i , 1)Ut0(j)] /Lt

+ [Dp↵(j , ↵)F (i , ↵)�D

p↵(j , 1)Fx0(i)] /Lx

= [Dp⌧ (i , ⌧)U(⌧, j)�D

p⌧ (i , 1)Ut0(j)] /Lt

+ [Dp↵(j , ↵)U(⌧1, ↵1)U(⌧2, ↵2)

⇥ �(i ,P(⌧1, ⌧2))�(↵,P(↵1, ↵2))] /(2L3/2x

L1/2t

)

� ⌫Dp↵(j , ↵) [D

p↵(↵,↵3)U(i , ↵3)

� Dp↵(↵, 1)U

x0(i)] /L2x

� Dp↵(j , 1)F

x0(i)/Lx

Its easier than it looks!


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Derived Types and Operator Overloading

The intimidating algebra of the previous slides and its myriadpotential for programming error are eliminated in Fortranwith the use of derived types and operator overloading.

uxw = intx(q1w(m),fa=q1aw(m),diff=.true.)

tauxw = nu*uxw

resw(1) = intt(q1w(m),fa=q10w(m),diff=.true.) &

+ intx(0.5_dp*q1w(m)**2-tauxw,fa=q1bw(m),diff=.true.)

I Each of the variable names ending in “w” above is oftype(walsh).

I Each walsh variable includes all wave components andtheir Jacobians with respect to the wave components ofevery dependent variable and boundary condition.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Steady Burgers Equation Convergence, ⌫ = 0.1

p = 4, n = 16

x

u

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Residual History

iteration

L1

0 10 20 30 4010-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100 2 3 4 5 6 7 8 9 10

Enrichment trigger

p

p = 9, n = 512

x

u

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Error Norm Historyp n L

p

1 L

p�11 /L

p

12 4 5.44 10�1

3 8 1.31 10�1 4.15

4 16 1.33 10�2 9.85

5 32 2.63 10�3 5.05

6 64 6.01 10�4 4.37

7 128 1.44 10�4 4.17

8 256 3.55 10�5 4.06

9 512 8.81 10�6 4.03

10 1024 2.19 10�6 4.02


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Steady Burgers Equation, ⌫ = 0.0001

p = 9, n = 512, under-resolved shock

x

u

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5 ν = 1. 10-4

L1

9= 3.92 10

-3

Detail, full series

x

u

-1 -0.5 0 0.5 10.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05 ν = 1. 10-4

L1

9= 3.92 10

-3

Detail, truncated series

x

u

-1 -0.5 0 0.5 10.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05 ν = 1. 10-4

L1

9t= 2.46 10

-4


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Run Times for Advection Test

u

t

+ u

x

= 0, u(x , 0) = u0(x), periodic boundaryProcessor 2.93 GHz Intel Core 2 Duo

p↵ p⌧ N↵N⌧ (N↵N⌧ )

2interior FWT boundary linear solve

3 3 64 4,096 1.8 10

�42.6 10

�41.0 10

�66.9 10

�5

4 4 256 65,536 1.9 10

�32.6 10

�33.4 10

�56.8 10

�4

5 5 1024 1,048,576 8.1 10

�26.2 10

�25.7 10

�41.6 10

�2

6 6 4096 16,777,216 2.8 10

�01.7 10

�09.8 10

�34.2 10

�1

7 6 8192 67,108,864 1.2 10

+16.5 10

�05.8 10

�22.5 10

�0

Problem is too big for laptop (4 GB 1067 MHz DDR3) ifp↵ + p⌧ > 13


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Riemann Problem - Sod Test Case

I Calorically perfect gas with � = 1.4 in a tube of length2 (�1 x 1) with a diaphragm at x = 0.

I Left state: p

L

= 1, ⇢L

= 1, u

L

= 0

I Right state: p

R

= .1, ⇢R

= .125, u

L

= 0

I Diaphragm breaks at t = 0.

I Shock moves to right with speed V

shock

= 1.75215.

I Contact discontinuity moves to right with speedV

contact

= 0.927452.

I Head of expansion fan moves to left with speedV

head

= �1.183216.

I Tail of expansion fan moves to left with speedV

tail

= �0.0702728.

I Waves reflect o↵ of end walls.


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Riemann Problem - Density

Density, p↵ = 7, p⌧ = 0, 8 subdomains in x

x

ρ

-0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

Walsh Series SolutionExact Solution


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Near Term Goals

I Document and release Walsh function module includingderived types and operator overloading.

I Complete multi-dimensional Riemann problem analysis.

I Replace Hyp-FUN3D reconstruction with Walshfunction algorithm. (HEDL support to address issues ofheating solution quality on tetrahedral grids.)


Peter A. Gno↵o



Fractal derivation








Burger’s Equation

Advection run times


Future work

Observations and Speculations

I Taylor series expansions are built on a requirement oflocal continuity.

I Newton relaxation and the Jacobian derive from C1

continuity assumptions in the Taylor series.

I Are new, useful insights and algorithms possible if werevisit classical derivations based on assumptions oflocal continuity (Taylor series) expansions and replacethem with Walsh series expansions requiring globalsupport?

o introduction to walsh functions in numerical...

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