REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY) 22 June 2015

2. REPORT TYPEBriefing Charts

3. DATES COVERED (From - To) 19 June 2015 – 22 June 2015

4. TITLE AND SUBTITLE Optimal Runge-Kutta Schemes for High-order Spatial and Temporal Discretizations

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) Mundis, N., Edoh, A. and Sankaran, V.

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER Q12J

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NO.

Air Force Research Laboratory (AFMC) AFRL/RQR 5 Pollux Drive Edwards AFB, CA 93524-7048

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) Air Force Research Laboratory (AFMC) AFRL/RQR 11. SPONSOR/MONITOR’S REPORT

5 Pollux Drive NUMBER(S)

Edwards AFB, CA 93524-7048 AFRL-RQ-ED-VG-2015-268

12. DISTRIBUTION / AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES For presentation at 22nd AIAA Computational Fluid Dynamics Conference; Dallas, TX; 22 June 2015 PA Case Number: #15352; Clearance Date: 6/29/2015

14. ABSTRACT Viewgraphs/Briefing Charts

15. SUBJECT TERMS N/A

16. SECURITY CLASSIFICATION OF:

17. LIMITATION OF ABSTRACT

18. NUMBER OF PAGES

19a. NAME OF RESPONSIBLE PERSON

V. Sankaran

a. REPORT Unclassified

b. ABSTRACT Unclassified

c. THIS PAGE Unclassified

SAR 29 19b. TELEPHONE NO

(include area code)

N/A Standard Form

298 (Rev. 8-98) Prescribed by ANSI Std. 239.18

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Op

tim

al R

un

ge-

Ku

tta

Sch

emes

fo

r H

igh

-ord

er

Sp

atia

l an

d T

emp

ora

l Dis

cret

izat

ion

s

2015

AIA

A S

ciT

ech

June

23,

201

5

Nat

han

L. M

un

dis

– E

RC

, In

c.

Aya

bo

e K

. Ed

oh

– U

CL

A

Ven

keS

anka

ran

– A

FR

L/R

Q

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Ou

tlin

e

• In

tro

du

ctio

n

•G

ove

rnin

g E

qu

atio

ns

–S

patia

l Dis

cret

izat

ions

–T

empo

ral D

iscr

etiz

atio

ns

•V

on

Neu

man

n A

nal

ysis

(V

NA

) •

Co

mp

uta

tio

nal

Res

ult

s –

One

-dim

ensi

onal

Wav

e

–T

hree

-dim

ensi

onal

Vor

tex

•C

on

clu

sio

ns

and

Fu

ture

Wo

rk

2

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Intr

od

uct

ion

• 

Hig

h-o

rder

in s

pac

e is

no

w c

om

mo

np

lace

•H

igh

-ord

er in

tim

e n

ot

so m

uch

• Is

th

is s

uff

icie

nt?

Is h

igh

-ord

er in

tim

e n

eed

ed?

•Li

miti

ng F

act:

Th

ere

are

no

A-s

tab

le b

ackw

ard

-dif

fere

nce

fo

rmu

la (

BD

F)

met

ho

ds

wit

h >

2nd

-ord

er a

ccu

racy

•T

hu

s, m

ult

ista

ge

met

ho

ds,

like

Ru

ng

e-K

utt

a (R

K)

met

ho

ds,

mu

st b

e u

sed

fo

r 3r

d - a

nd

hig

her

-ord

er

•E

xplic

it R

K m

eth

od

s ar

e n

ot

amen

able

to

sti

ff p

rob

lem

s

Obj

ectiv

e:T

o f

ind

op

tim

al d

iag

on

ally

-im

plic

it

Ru

ng

e-K

utt

a ti

me

inte

gra

tors

fo

r u

se w

ith

h

igh

-ord

er s

pat

ial d

iscr

etiz

atio

ns

3

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Go

vern

ing

Eq

uat

ion

s

• D

ual

Tim

e S

tep

pin

g:

•Q

uas

i-lin

ear

Fo

rm:

•R

esid

ual

Fo

rm:

4

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Sp

atia

l Dis

cret

izat

ion

s

• C

entr

al D

iffe

ren

ces

wit

h a

dd

ed a

rtif

icia

l dis

sip

atio

n

•C

entr

al d

iffe

ren

ces:

•S

cala

r ar

tifi

cial

dis

sip

atio

n:

5

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Tem

po

ral D

iscr

etiz

atio

ns

• R

un

ge-

Ku

tta

Met

ho

ds:

6

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

ES

DIR

K M

eth

od

s

• E

xplic

it fir

st s

tage

Sin

gly-

Dia

gona

lly

I mpl

icit

Run

ge-K

utta

–S

tiffly

acc

urat

e

–S

econ

d-or

der

stag

e ac

cura

cy

–F

SA

L –

Firs

t is

the

Sam

e A

s L

ast

7

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

ES

DIR

K3

and

4

8

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

ES

DIR

K5

9

The

sin

gle

bigg

est d

raw

back

of u

sing

th

ese

sche

mes

is ty

ping

them

out

!

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Vo

n N

eum

ann

An

alys

is

• O

ften

use

d t

o s

tud

y st

abili

ty o

f sc

hem

es

• V

on

Neu

man

n a

nal

ysis

is u

sed

to

co

mp

are

sch

emes

fo

r ac

cura

cy

–D

issi

patio

n er

ror

–D

ispe

rsio

n er

ror

•A

ssu

mes

lin

ear,

per

iod

ic p

rob

lem

s

•V

NA

th

eory

an

d m

ore

res

ult

s ar

e in

th

e as

soci

ated

pap

er

10

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Dis

sip

atio

n, C

FL =

1.0

11

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Dis

per

sio

n, C

FL =

1.0

12

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Dis

sip

atio

n, C

FL =

10.

0

13

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Dis

per

sio

n, C

FL =

10.

0

14

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

1-D

Aco

ust

ic W

ave

• U

np

ertu

rbed

Mac

h n

um

ber

of

0.5

•P

ertu

rbat

ion

wav

e -

20 p

oin

ts p

er w

ave

reso

luti

on

• M

ore

res

ult

s in

th

e p

aper

15

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

1-D

, CFL

= 1

.0, 1

0 P

erio

ds

16

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

1-D

, CFL

= 1

0.0,

1 P

erio

d

17

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

1-D

, CFL

= 1

.0, 1

000

Per

iod

s

18

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

3-D

Isen

tro

pic

Vo

rtex

• 

Fre

e-st

ream

Mac

h n

um

ber

of

0.5

• P

ertu

rbat

ion

- 1

1 p

oin

ts a

cro

ss t

he

vort

ex

• M

ore

res

ult

s in

th

e p

aper

19

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

3-D

, CFL

= 1

.0, 4

0 L

eng

ths,

11

Po

ints

Acr

oss

th

e V

ort

ex

20

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

3-D

, CFL

= 1

.0

Dif

fere

nt

Res

olu

tio

ns

21

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

3-D

, CFL

= 8

.0, 4

0 L

eng

ths,

11

Po

ints

Acr

oss

th

e V

ort

ex

22

Alm

ost 1

vor

tex

wid

th d

own

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Sn

eak

Pea

k: F

ilter

ing

23

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Co

ncl

usi

on

s

• 2n

d - a

nd 3

rd-o

rder

tim

e in

teg

rato

rs f

or

5th -

ord

er

spat

ial s

chem

es a

re in

adeq

uat

e –

The

sam

e or

der

of s

patia

l and

tem

pora

l dis

cret

izat

ions

is p

refe

rabl

e

–H

owev

er, E

SD

IRK

5 is

not

muc

h be

tter

than

ES

DIR

K4

•7

impl

icit

stag

es v

s. 5

impl

icit

stag

es

•H

igh

er-o

rder

tim

e in

teg

rato

rs:

–D

o no

t sho

w s

igni

fican

t im

prov

emen

t on

coar

se g

rids

at C

FL

of o

ne

–A

re b

ette

r at

hig

h C

FL

num

ber

–A

re b

ette

r on

hig

hly

refin

ed g

rids

•S

pat

ial e

rro

r u

sual

ly d

om

inat

es f

or

typ

ical

C

FL n

um

ber

s an

d g

rid

res

olu

tio

ns

–C

entr

al d

iffer

ence

plu

s ar

tific

ial d

issi

patio

n sc

hem

es a

re in

adeq

uate

24

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Fu

ture

Wo

rk

• Im

ple

men

t m

ore

acc

ura

te s

pat

ial s

chem

es o

f th

e sa

me

ord

ers

of

accu

racy

–

Com

pact

-diff

eren

ce s

chem

es

–F

ilter

ing

sche

mes

•D

eriv

e b

ette

r E

SD

IRK

sch

emes

tai

lore

d t

o t

he

des

ired

dis

sip

atio

n a

nd

dis

per

sio

n p

rop

erti

es

•A

dd

pre

con

dit

ion

ing

to

tak

e m

axim

um

ad

van

tag

e o

f th

e E

SD

IRK

tim

e in

teg

rato

rs f

or

stif

f p

rob

lem

s –

Impr

oved

con

verg

ence

effi

cien

cy

–Im

prov

ed s

olut

ion

accu

racy

25

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Qu

esti

on

s??

?

26

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

tr

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

Ext

ra S

lides

27

Dis

trib

uti

on

A –

Ap

pro

ved

fo

r p

ub

lic r

elea

se;

Dis

trib

uti

on

Un

limit

ed

3-D

, CFL

= 8

.0

Dif

fere

nt

Res

olu

tio

ns

28