CDn.1 n-Dimensional Convection-Diffusion Unsteady n-D convective transport of a scalar q(x, t)

Interpretation of q , Pa, Pb is problem-specific

2

0 0 0

1( ) 0 , onPa

( ) Pb( ) 0 , on

( , ) ( , ) , on( , ) ( ) , on

ref n R

b b b D

qq q q s tt

q q q q f t

q t q t tq t q t

∂= + ⋅∇ − ∇ − = Ω×

∂= ∇ ⋅ + − + = ∂Ω ×

= ∂Ω ×= Ω ∪ ∂Ω×

u

n

x xx x

L

u(x,t) n-dimensional fluid velocity field s(x,t), fn distributed source, boundary flux Robin/Dirichlet BCs may be time-dependent

CDn.2 GWSh Ingredients for n-D L(q) with BCs

FE implementation of GWSh and/or TWSh

( ){ } { }

{ }β

: ( ) ( ) 0: ( , ) ( , ) ( , ) ( , )

: ( , ) ζ,η ( )

: GWS ( ) ( )dτ 0 TWS

: TWS TS [JAC]{ } RES

[

e

v

N he e

Te k e

N m N h

h

PDE system q q t s BCs ICapproximation q t q t q t q t

FE basis q t N Q t

error exremization q

matrix statement Q tΩ

= ∂ ∂ + ∇⋅ − − = + +

≈ ≡ ≡ ∪

≡

= Ψ ≡ ⇒

+ θ ⇒ Δ = −Δ

∫

f fx x x x

x

x

L

L

[ ]( ) ( ){ } [ ]( ){ } { }

{ } { }22 ( )

0L2

1

JAC] S JAC , {RES} S {RES}

RES UVWVEL [DIFF] [HBC] ( )

[JAC] [MASS] RES /

: ( ) C data C

γ min(1; , 1 Pa 0): & ( , , , , , , ,

e e ee

e ee e e e

e e e e

h fe t EE

Q b

t Q

asymptotic convergence e t h t q

k r forerror spectra U G f k h t

γ θ

−

ω

= =

= + + − ⋅

= + θΔ ∂ ∂

≤ + Δ

= − >

⇒ ω Δ θ α β γ)

CDn.3 n-Dimensional GWSh, FE Bases, ODE Form

For t ime-dependence in coefficient set {Q( t)} • a l l n -d imens iona l bases {N k (ζ ,η )} remain as es tab l i shed • GWSN, hence GWSh, TWSh always yield large (!) ODE systems Galerkin weak statement for L (q) + BCs + IC

GWSh + θTS produces algebraic form

[JAC]{ΔQ} = - Δt{RES}n with {Q}n+1 = {Q}n + {ΔQ}

( )

{ }0}RES{}]{MASS[

}b{}{]HBC[]DIFF[]UVWVEL[d

}{d]MASS[S

}WS{SGWS}0{dτ)()(GWS

'

β

≡+≡

⎟⎠⎞

⎜⎝⎛ −+++=

=⇒≡Ψ≡ ∫Ω

Q

Qt

Q

q

eeeeee

ee

eehNN Lx

CDn.4 GWSh + θTS Statement Contributions

G W S h + θ T S fo r m e d b y a s s e m b ly o v e r Ω e

n e w m a t r i x d u e t o f l u i d c o n v e c t i o n recall e l e m e n t m a t r i x n o t a t i o n c o n v e n t i o n

[M …]e , M ⇒ (A, B, C) for 1 ≤ n ≤ 3

Se( [JAC]e){ΔQ} = -Δt Se({RES}en)

[JAC]e = [M200]e + θ Δt([UVWVEL]e + [DIFF]e + [HBC]e)

{RES}en = ([UVWVEL]e + [DIFF]e + [HBC]e){Q}e

n –{b}e

{b(t)}e = ([M200]e{SRC}e + [HBC]e{QR}e - [N200]e{FN}e)n+θ

nJJJ

wu

eTe

eeeeeee

≤≤⇒

++=

1,]300M[}U{

]203M[]202M[]201M[]UVWVEL[ ν

CDn.5 Fluid Convection FE Matrix Construction

Fluid convection on n-D is sole new matrix

Additional template contributions (TDT = θΔt )

1,1,1,}]{300M[}U{ET

}{dηdetηη

}{}}{{}U{ˆ

+≤≤≤≤=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

= ∫Ω

nnKnJforQKJKJ

Qx

NNNJ

eTee

ee

ej

k

k

TTe

e

]][3003C)[0;708090}(3U2U1U){)(TDT(]][3002C)[0;405060}(3U2U1U){)(TDT(

]][3001C)[0;102030}(3U2U1U){)(TDT(]JAC[

++++++++=eU

0,}{]JAC)[/1(}RES{}RES{ >θθ+Δ⇒Δ eeene QUtt

dτ}{}{]UVWVEL[j

h

jee xquNQ

e ∂∂

≡ ∫Ω

CDn.6 Accuracy, Convergence & Stability

A sym ptot ic convergence error es t im ates rem ain

Stab i l i ty contro ls accuracy for large P a , reca l l C D 1

C = 0.5 θ = 0.5

Amplification Factor Damping

Artificial diffusion spectra Phase velocity error spectra

Ef

t

t

teE

h

Ef

teE

h

qtqhte

rkqthte

k 0)(2

H21-

0)(2

2L21-

Cdτ),(C)(:0Pa

)1,min(γ,CdataC)(:0Pa

01

θ

θ

Δ+≤=

−=Δ+≤>

∫ +τ

γ

x

CDn.7 GWSh + θTS Newton Template, n = 3

Problem statement L ( q ) = ∂ q ∂ t

+ u ⋅ ∇ q - 1 Pa

∇ 2 q - s = 0

( q ) = ∇ q ⋅ n + Pb q - q ref + f n = 0

Template pseudo-code for {FQ}e for {Nk+} basis, n = 3

Newton ( [MASS] + θ Δt ([UVWVEL] + [DIFF] + [HBC] )){δQ}p+1 = -{FQ}p

{FQ} = [MASS]{QP – QN} + Δt ( [UVWVEL] + [DIFF] + [HBC]{Q} – {b})n+θ

{FQ}e = ( )( ){ }(0; 1)[C200]{QP} +( - )( ){ }(0; 1)[C200]{QN} +( )( ){U1 + U2 + U3}(102030; 0)[C3001]{QP} +( )( ){U1 + U2 + U3}(405060; 0)[C3002]{QP} +( )( ){U1 + U2 + U3}(708090; 0)[C3003]{QP} +(PAI)( ){ }(112233; -1)[C211]{QP} +(PAI)( ){ }(475869; -1)[C223]{QP} +(PAI)( ){ }(445566; -1)[C222]{QP} +(PAI)( ){ }(475869; -1)[C232]{QP} +(PAI)( ){ }(778899; -1)[C233]{QP} +(PAI)( ){ }(172839; -1)[C213]{QP} +(PAI)( ){ }(142536; -1)[C221]{QP} +(PAI)( ){ }(172839; -1)[C231]{QP} +(PAI)( ){ }(142536; -1)[C212]{QP} +( - )( ){ }(0; 1)[C200]{SRC} +(PB, PAI)( ){ }(0; 1)[B200]{QP} +(- PB, PAI)( ){ }(0; 1)[B200]{QR} +(PAI)( ){ }(0; 1)[B200]{FN}

(omitting TDT, TDT1, P, N duplications)

CDn.8 GWSh + θTS Linear Algebra Issues

GWSh + θTS produces large (!) matrix statement Newton j acob ian Linear algebra exchanges speed for slower convergence

where : N, M a re “ i t e ra t ion ma t r i ces , ” p i s i t e ra t ion index

[JAC]{ΔQ} = -Δt{RES} [JAC]{δQ}p+1

= -{F}p

{ } { } { }...)]HBC[]DIFF[]UVWVEL([]MASS[

/}RES{]MASS[/F]JAC[+++Δθ+=

∂∂Δθ+=∂∂≡t

QtQ

bMxNxxxbAfx

bAxbAx

pppp

pppp

+=

=

=

=

−

−−

−

1

21

1

:iterationlinear.)..,,,,(:iterationstationary

:rulesCramer':notationmatrixstandard

CDn.9 Linear Point Iterative Methods

L i n e a r i t e r a t i o n x p = N p x p -1 + M pb , fo r A x = b ite ra te e rro r:

ite ra tiv e co n v erg en ce :

d e p e n d s s t r i c t l y o n i t e r a t i o n m a t r i x N a n d i n i t i a l e r r o r e 0 P o i n t i t e r a t i v e m e t h o d s

1

11

1

−

−−

−

=

−+=

−=

−≡

pp

ppp

p

pp

eNbAbMxN

bAxxxe

011

211

...... eNNN

eNNeNe

pp

pppppp

−

−−−

=

⋅⋅

==

p a rtitio n m atrix : A = L + D + U req u ire n o m atrix so lv es : N ≡ N (L , D , U )

L , D , U are sq u are m atrices , sam e e lem en ts as A D ⇒ d iag o n a l L , U ⇒ tr ian g u la r ( lo w er, u p p er)

CDn.10 Familiar Linear Point Iterative Methods

P i c a r d ( J a c o b i ) i t e r a t i o n G a u s s -S e id e l i t e r a t io n S u c c e s s iv e O v e r -R e la x a t io n ( S O R )

ii

pp

pp

dDDMMULDNN

bDxULDx

/1),(:hence

)(

1

11

111

=

=⇒+−=⇒

++−=

−

−−

−−−

bDUxDLxDxDLMUDLN

bDLUxDLx

ppp

pp

1111

11

111

:formalgebraic)(,)(:hence

)()(

−−−−

−−

−−−

+−−=

+=+−=

+++−=

( )2ω1:factorrelaxation

ω)ω)ω1(()ω( 11

≤≤+−−+= −− bxUDLDx pp

CDn.11 Approximate Factorization

GWS h + θTS matrix statement [JAC] {ΔQ} = -Δt {RES} Newton jacobian: [JAC] = [MASS] + θΔt ([UVWVEL] + [DIFF] + [HBC]) Approximate factorizat ion {P s }and {S s } a re ro w -co lu mn “sh u f f led ”

{ }

{ }s

s

SQ

PStP

≡Δ

≡Δ−=

}~]{J3[

}{}]{J2[RES}]{J1[:sequencesolution

partition [JAC]: [JAC] ≅ [J1]{J2][J3] matrix statement: [J1][J2][J3]{ΔQ}=-Δt{RES}

CDn.12 Tensor Product Approximate Factorization

GWSh + θTS jacobian, n = 3 on Ωe [JAC]e = [C200]e + θΔt (uk [C20K]e + Pa-1[C2KK]e + [HBC]e) Tensor (outer) matrix product (⊗)

then: A200 e ⊗ A200 e =

ly

6 2 1

1 2 ⊗ l x

6 2 1

1 2

= l x l y

36 4 2 1 2 2 4 2 1 1 2 4 2 2 1 2 4

= B200 e

A200 e ⊗ u 1 A201 e =

ly

6 2 1

1 2 ⊗ U x

2 -1 1 -1 1

= l y U x

12 -2 2 1 -1 -2 2 1 -1 -1 1 2 -2 -1 1 2 -2

= U 1 B201 e

also:

eeee ]200C[]200A[]200A[]200A[:then =⊗⊗

and: [A200]e ⊗ [A211]e = [B211]e

for n = 1 and {N1}: [MASS]e ⇒ [A200]e

CDn.13 Tensor Matrix Products, Error

N ew ton jacob ian [JAC] = [C200] + θΔt (uk[C20K] + Pe-1[C2KK] + [HBC]) [JAC]e ≅ [J1]e [J2]e [J3]e [JK]e = [A200]e + θΔt (uK[A20K]e + Pe-1[A2KK]e + [HBC]) T ensor m atr ix product o f the sum f a c to r i z a t i o n e r r o r m a t r ix g e n e r a t e d v i a

[ J 1 ] e ⊗ [ J 2 ] e = ([MX] + θΔt [CX + DX + HX])e ⊗ ([MY] + θΔt [CY + DY + HY])e

= [MX] ⊗ [MY] + [MX] ⊗ θΔt [CY + DY + HY]

+ [MY] ⊗ θΔt [CX + DX + HX]

+ (θΔt)2 [CX + . . . ] ⊗ [CY + . . . ]

= [B200]e + θΔt [B201 + B202 + B211+ B222 + HBC]e

+ (θΔt)2 [error matrix]

CDn.14 Time-Splitting, Alternating Direction Implicit

T e n s o r p r o d u c t A F a l g o r i t h m T i m e - s p l i t t i n g a l g o r i t h m [JX] {ΔQ}n+1/2 = - Δt {RESX(Qn)} {Q}n+1/2 = {Q}n + {ΔQ}n+1/2 [JY] {ΔQ}n+1 = - Δt {RESY(Qn+1/2)} {Q}n+1 = {Q}n+1/2 + {ΔQ}n+1

A l t e r n a t i n g d i r e c t i o n i m p l i c i t ( A D I ) [JX] {ΔQ}n+1/2 = - Δt {RESY(Qn)} {Q}n+1/2 = {Q}n + {ΔQ}n+1/2 [JY] {ΔQ}n+1 = - Δt {RESX(Qn+1/2)} {Q}n+1 = {Q}n + {ΔQ}n+1/2 + {ΔQ}n+1

{RES}toionsapproximatno:note}~{}{}{

}RES{}~]{3J][2J][1J[

1 QQQ

tQ

nn Δ+=

Δ−=Δ

+

CDn.15 GWSh + θTS Jacobian Templates, n = 3

Newton jacobian Tensor product jacobians [JAC]e = ( )( ){ }(0; 1)[C200][ ] [J1]e = ( )( ){ }( )[A200][det] + (TDT)( ){U1 + U2 + U3}(102030; 0)[C3001][ ] + (TDT)( ){U1 + U2 + U3}(102030; 0)[A3001][ ] + (TDT)( ){U1 + U2 + U3}(405060; 0)[C3002][ ] + (PEI, TDT)( ){ }(114477; -1)[A211][ ] + (TDT)( ){U1 + U2 + U3}(708090; 0)[C3003][ ] + (NU, PEI, TDT)( ){ }(0;1)[A200][ ] + (PEI, TDT)( ){ }(112233; -1)[C211][ ] + (PEI, TDT)( ){ }(445566; -1)[C222][ ] [J2]e = ( )( ){ }( )[A200][det] + (PEI, TDT)( ){ }(778899; -1)[C233][ ] + (TDT)( ){U1 + U2 + U3}(405060; 0)[A3001][ ] + (PEI, TDT)( ){ }(142536; -1)[C221][ ] + (PEI, TDT)( ){ }(225588; -1)[A211][ ] + (PEI, TDT)( ){ }(142536; -1)[C212][ ] + (NU, PEI, TDT)( ){ }(0;1)[A200][ ] + (PEI, TDT)( ){ }(475869; -1)[C223][ ] + (PEI, TDT)( ){ }(475869; -1)[C232][ ] [J3]e = ( )( ){ }( )[A200][det] + (PEI, TDT)( ){ }(172839; -1)[C213][ ] = (TDT)( ){U1 + U2 + U3}(708090; 0)[A3001][ ] + (PEI, TDT)( ){ }(172839; -1)[C231][ ] + (PEI, TDT)( ){ }(336699; -1)[A211][ ] + (NU, PEI, TDT)( ){ }(0; 1)[B200][ ] + (NU, PEI, TDT)( ){ }(0;1)[A200][ ]

CDn.16 Verification, Benchmarks & Validation

Protocol for simulation quality assessment Results of these “computational experiments" provide TWSh + θTS templates support full spectrum assessments

• verification - compare simulation to known analytical solutionsPeclet problem traveling mass packet rotating cone

• benchmark - compare simulation to quality numerical results Gaussian plume

• validation - compare simulation to quality experimental data genuine geometries realistic diffusion models

• error mechanism assessments, mesh dependence• knowledge of limitations of theory/code • confidence in the simulation methodology

CDn.17 Rotating Cone, n = 2 Verification

Initial condition & exact solution

Crank-Nicolson FD GWSh k = 1 FE

TWSh, k = 1 FE, α = 0, β = 0, γ = -0.5

u

0 0 0

on : ( ) 0on : ( ) 0

ˆdata : ω: ( , ) ( ) "gaussian "

t

in b b

q q qq qr

at t q t qθ

Ω = + ⋅∇ =∂Ω = ≡

== ⇒

ux

u ex x

L

θθ ==+ 0.5Csolution,TSWSVarious h

CDn.18 Gaussian Plume, n = 2, 3 Benchmarks

GWSh steady solution, n = 2

1 1 2 2 3 -1 1 1 2 3 5 5 6 8 9 9 10 4 6 9 12 14 16 18 20 21 22 24 25 25 26 2 12 21 28 35 38 41 44 45 46 47 48 48 48 49 49 49 49 0 8 50 93 100 97 96 93 90 88 86 83 81 79 77 75 74 72 71 69 0 16 92 159 156 142 132 123 116 110 105 100 97 94 90 87 85 82 80 78

0 25 98 162 166 145 137 129 121 115 109 105 100 97 93 90 88 85 83 81

-2 10 94 169 164 148 138 129 122 116 111 106 101 98 95 91 89 86 83 81 -1 4 50 97 103 101 98 95 93 90 88 86 83 81 80 77 76 74 73 71 2 10 19 26 31 35 38 41 43 44 45 46 47 47 48 48 48 48 -2 -2 -1 1 3 6 9 11 14 15 17 19 20 22 23 24 25 1 2 2 4 5 5 7 8 8 9 1 1 2 2

u

U

1 3 10 26 48 71 81

81 71 48 26 10 3 1

Gaussian

FE k = 1

FE k = 2 source region

1on : ( ) Pa 0ˆon : 0, 0

data : , ( , , ), ( ) givenin out

x y z

q q k q sq q

k k k k s

−Ω = ⋅∇ − ∇ ⋅ ∇ − =∂Ω = ⋅∇ =

= = =

un

u i x

L

GWSh steady-state DOF solutions, n = 2

kji ˆˆˆ0,20Pa ++== k

CDn.19 Enclosure Flow, n = 3 Validation

velocity, Re = 15,000, Ret = 14 temperature, Gr / Re2 = 4.3

on data∂Ω: " ,"q qin out= ⋅∇ =n 0

2

0 0

(1 Re ) Gr ˆon : ( ) 0Re Re

ˆon : ( , ) { , , , } given , 0at : ( , ) 0

t

jj j

Tin out

q qq u qt x x

q q t u v qt q t

⎛ ⎞∂ ∂ + ∂Ω = + − + Θ =⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

∂Ω = = ω Θ ⋅ ∇ ==

g

x nx

L

Experiment, mid-plane flow speed

GWSh + θTS k = 1 FE solution

CDn.20 Mass Transport Experiments on n = 3

GWSh, k=1, Ret=14, Sct=1 TWSh, k=1, β=0.9, Sct=1 GWSh, k=1, Sct=0.1 TWSh, Sct=1, RCN

CDn.21 TWSh + θTS Summary

Essential ingredients of TWSh + θTS for all n Template pseudo-code converts theory to practice

{ } { }{ } { }

1: ( ) 1 Pa 0, BCs ICPa

: ( , ) , ( )

: GWS TWS TS [JAC] RES

: [JAC] [JAC approx] matrix iteratio

tj

j j j

TN he e e k

h hn

q q qPDE system q u st x x x

approximation q t q q q q N Q t

minimize error Q t

linear algebra

asym

⎛ ⎞∂ ∂ ∂ ∂= + − + − = + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

≅ ≡ = ∪ =

⇒ + θ ⇒ Δ = −Δ

⇒ ⇒

x

L

: , , , Pa, Pa , data

: , (ω, , , , ,α,β, γ)

h h t

Eptotic convergence e f k r

error spectra U G f k h tω

⎛ ⎞≤ Ω⎜ ⎟⎝ ⎠

⇒ Δ θ

- “code in English” - readily input physics nuances - precise non-linearities - linear algebra options