Numerical Methods in Heat Mass Momentum Transfer (Lecture Notes)JayathiMurthy

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ME 608

Numerical Methods for Heat, Mass

and Momentum Transfer

Jay

Jayathathii Y. Y. MurMurthythy

Professor, School of Mechanical Engineering Professor, School of Mechanical Engineering

Purdue University Purdue University jmurthy@ecn.purdue.edu jmurthy@ecn.purdue.edu Spring 2006 Spring 2006

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Lecture 1: Introduction to ME 608

Conservation Equations

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Outline of Lecture



 Course organizationCourse organization 

 Introduction to CFDIntroduction to CFD 

 Conservation equations, general scalar transport equationConservation equations, general scalar transport equation 

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Motivation

Huge variety of industrial flows: Huge variety of industrial flows: •Rotating machinery •Rotating machinery •Compressible/incompressible •Compressible/incompressible aerodynamics aerodynamics •Manifolds, piping •Manifolds, piping •Extrusion, mixing •Extrusion, mixing

•Reacting flows, combustion …. •Reacting flows, combustion …. Impossible to solve Navier-Stokes Impossible to solve Navier-Stokes equations

equations analytically analytically for for thesethese applications!

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History



 EarliEarliest “Cest “CFD”FD” work by work by L.F. RiL.F. Richardchardsonson

(1910) (1910)

»

» UsUseed hd huumaman cn comompuputetersrs »

» ItItereratativive soe solulutitionons os of Laf Laplplacace’e’ss eqeqnn using finite-difference methods, flow using finite-difference methods, flow over cylinder etc.

over cylinder etc. »

» ErErroror esr estitimamatetes, es, extxtrarapopolalatition ton too zero error

zero error

“So far I have paid

“So far I have paid piece rates for thepiece rates for the operation (Laplacian) of about n/18 operation (Laplacian) of about n/18 pence per coordinate point, n being the pence per coordinate point, n being the numbe

number of digr of digits …its … one of tone of the quihe quickesckestt boys

boys averaged averaged 2000 2000 operationsoperations (Laplacian)

(Laplacian) per week per week for numbfor numbers of ers of 33 digits, those done wrong being

digits, those done wrong being discounted …”

discounted …”

Richardson, 1910 Richardson, 1910

Also researched mathematical Also researched mathematical models for causes of war : models for causes of war : Generalized Foreign Politics Generalized Foreign Politics (1939)

(1939)

Arms and Insecurity(1949) Arms and Insecurity(1949) Statistics of Deadly Quarrels Statistics of Deadly Quarrels (1950) (1950) Lewis F. Richardson Lewis F. Richardson (1881-1953) (1881-1953)

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History



 Relaxation methods (1920’s-50’s)Relaxation methods (1920’s-50’s) 

 LandmLandmark papark paper by Courer by Courant, Friant, Friedricedrichshs andand

Lewy

Lewy for hypfor hyperbolerbolic equaic equationtions (1928s (1928))



 Von Neumann stability criteria for parabolicVon Neumann stability criteria for parabolic

problems (1950) problems (1950)



 HarloHarlow and Fromw and Frommm (1963(1963) compu) computed unsted unsteadyteady

vortex street using a

vortex street using a digital computer.digital computer.



 They published a Scientific American articleThey published a Scientific American article

(1965) which ignited interest in modern CFD (1965) which ignited interest in modern CFD and the idea of computer experiments

and the idea of computer experiments



 Boundary-layer codes developed in the 1960-Boundary-layer codes developed in the

1960-1970’s (G

1970’s (GENMIX by PENMIX by Patankatankarar and Spaland Spalding inding in 1972 for eg.)

1972 for eg.)



 Solution techniques for incompressible flowsSolution techniques for incompressible flows

published through the 1970’s (SIMPLE family of published through the 1970’s (SIMPLE family of algor

algorithms bithms by Patanky Patankarar and Spaand Spalding folding for eg.)r eg.)



 Jameson Jameson computed computed Euler Euler flow flow over cover completeomplete

aircraft (1981) aircraft (1981)



 Unstructured mesh methods developed inUnstructured mesh methods developed in

1990’s 1990’s John John von Neumann von Neumann (1903-1957) (1903-1957) Richard Courant Richard Courant (1888-1972) (1888-1972)

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Conservation Equations



 Nearly all physical processes of interest to us Nearly all physical processes of interest to us are governed byare governed by

conservation equations conservation equations

»

» MasMass, s, mommomententum um eneenergy rgy conconserservatvationion



 WrittWritten in terms of speen in terms of specificcific quantiquantities (per unities (per unit mass basist mass basis))

»

» MMoommeennttuum m ppeer r uunniit t mmaasss s ((vveelloocciittyy)) »

» EnEnerergy gy pper er ununit it mmasass es e



 Consider a specific quantityConsider a specific quantity

φφ

»

» CouCould be mold be momenmentum ptum per uner unit mait mass, ess, enernergy pegy per unir unit mast mass..s..



 Write conservation statement forWrite conservation statement for

φφ

for control volume of sizefor control volume of size

∆

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Conservation Equations (cont’d)

Accumulation of

φφ

in in control control volume volume over over time time stepstep

∆

t =t = Net influx of

Net influx of

φφ

into control volumeinto control volume -- NeNet et efffflulux ox off

φφ

out of control volumeout of control volume + Net generation of

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Conservation Equations (cont’d)

Accumulation: Accumulation:

Generation: Generation:

Influx and Efflux: Influx and Efflux:

(10)

Diffusion and Convection Fluxes

Diffusion Flux

Convection

Flux

Net flux

Velocity Vector

Diffusion coefficient

Γ Γ

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Combining…

Taking limit as

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General Scalar Transport Equation

Or, in vector form:

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Conservation Form

Consider steady state. The conservation form of the

scalar transport equation is:

Non-Conservation Form

Finite volume methods always start wit

Finite volume methods always start with the conservation

h the conservation

form

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General Scalar Transport Equation

Storage

Storage Convection Convection Diffusion Diffusion Generation Generation

Recall:

φφ

is a is a specific specific quantity quantity (energy per (energy per unit masunit masss say)

say)

V : velocity vector V : velocity vector

Γ

: Diffusion coefficient: Diffusion coefficient

ρρ

: density: density

S: Source term (Generation per unit volume W/m S: Source term (Generation per unit volume W/m33₎₎

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Continuity Equation

0 0 )) ((

==

⋅⋅

∇

++

∂∂

V V

ρ

t t

Here,

φφ

= 1

Γ

= 0

S = 0

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Energy Equation

h = sensible enthalpy per unit mass, J/kg h = sensible enthalpy per unit mass, J/kg k = thermal conductivity

k = thermal conductivity S

S_h_h = energy generation W/m= energy generation W/m33

Note:

Note: h ih in conn convection vection and sand storage torage termsterms T in diffusion terms

T in diffusion terms How to cast in the form

How to cast in the form of the general scalar transportof the general scalar transport equation?

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Energy Equation (cont’d)

Equation of State

Substitute to Find

Here,

φφ

= h

Γ

= k/C

_p_p

S = S

_h_h

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Momentum Equation

X-Momentum Equation

j j ii ij ij j j ii

u

x

τ

=

µ

⎛

⎜

_⎜

∂∂

++

∂∂

⎞⎞

⎟⎟

_⎟⎟

∂

∂∂

⎝

⎠⎠

Here,

φφ

= u

Γ

=

µ µ

S = S

_u_u

-

p

x

∂∂

S is goo

S is good “dumd “dumpinping grog groundund”” forfor everything

everything that that doesn’t doesn’t fit fit intointo the other terms

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Species Transport Equation

Y

Y_ii = kg of specie i /kg of mixture= kg of specie i /kg of mixture

Γ

_ii = diffusion coefficient of i in mixture i= diffusion coefficient of i in mixture i R

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Closure



 In this lecture weIn this lecture we

»

» DevDeveloeloped tped the prhe proceocedurdure for dee for develvelopioping the gng the goveovernirningng equation for the transport of a

equation for the transport of a scalarscalar

φφ

»

» ReRecocogngnizized ed ththe ce comommomonanalility ty of of trtrananspsporort ot off

–

– MasMass, mos, momenmentumtum, ener, energy, sgy, specpeciesies

»

» CastiCasting ang all tll these hese differdifferent ent equatiequations ons into into this this singlsingle fore form ism is very useful

very useful »

» Can dCan devievise a sinse a single mgle methethod to sood to solve thlve this clis class oass of govef governirningng equation

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Lecture 2: The General Scalar Transport

Equation

Overview of Numerical Methods

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Last time…



 Wrote conservation statement for a control volumeWrote conservation statement for a control volume 

 Derived a general scalar transport equationDerived a general scalar transport equation 

 Discovered that all transport processes commonalitiesDiscovered that all transport processes commonalities

» » SSttoorraaggee » » DDiiffffuussiioonn » » CCoonnvveeccttiioonn » » GGeenneerraattiioonn

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This time…



 Examine important classes of partial differential equations andExamine important classes of partial differential equations and

understand their behavior understand their behavior



 See how this knowledge applies to the general scalar transportSee how this knowledge applies to the general scalar transport

equation equation



 Start a general overview of the main elements of all numericalStart a general overview of the main elements of all numerical

methods methods

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General Scalar Transport Equation

Storage

Storage Convection Convection Diffusion Diffusion Generation Generation

Recall:

φφ

is a is a specific specific quantity quantity (energy per (energy per unit masunit masss say)

say)

V : velocity vector V : velocity vector

Γ

: Diffusion coefficient: Diffusion coefficient

ρρ

: density: density

S: Source term (Generation per unit volume W/m S: Source term (Generation per unit volume W/m33₎₎

(25)

Classification of PDEs

Consider

Consider the

the second-orde

second-order

r partial d

partial differential

ifferential equation

equation for

for

φφ

(x,y):

Co

Coef

effi

fici

cien

ents

ts a,

a,b,

b,c,

c,d,

d,e,

e,ff ar

are

e lin

linea

ear

r --

-- no

not

t fu

func

ncti

tion

ons

s of

of

φφ

, but can be functions of (x,y)

Discriminant

D

D <

< 0

0 Elliptic

Elliptic PDE

PDE

D=0

Parabolic

Parabolic PDE

PDE

D>0

(26)

Elliptic PDEs

Consider 1-D heat conduction in a Consider 1-D heat conduction in a plane wall with constant thermal plane wall with constant thermal conductivity conductivity Boundary conditions Boundary conditions Solution: Solution: T T _o_o T T _L_L

(27)

Elliptic PDE’s

T T _o_o T T _L_L •• T(x) is influenced by bothT(x) is influenced by both

boundaries boundaries

•• In thIn the abse absence ence of soof source urce termsterms,, T(x) is bounded by the values on T(x) is bounded by the values on both boundaries

both boundaries

•Can we devise numerical •Can we devise numerical

schemes which preserve these schemes which preserve these properties?

(28)

Parabolic PDEs

Consider 1D unsteady conduction Consider 1D unsteady conduction in a slab with constant properties: in a slab with constant properties:

Boundary and initial conditions

Boundary and initial conditions Solution:Solution:

T T ₀₀ T T _i_i T T ₀₀

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Pa

Para

rabo

bolic

lic PD

PDEs

Es (c

(con

ont’

t’d)

d)

T T ₀₀ T T _i_i T T ₀₀

•• The solution at T(x,t) The solution at T(x,t) is influenced by is influenced by the boundaries, justhe boundaries, just as with elliptit as with elliptic PDEsc PDEs •We need onl

•We need only initial y initial condtionscondtions T(x,0). We do T(x,0). We do not need future not need future conditionsconditions •Initial conditions only affect future conditions, not past conditions

•Initial conditions only affect future conditions, not past conditions •• Initial coInitial conditions affenditions affect all ct all spatial pspatial points in oints in the futurethe future

•• A steaA steady stady state is rete is reacheached as t->d as t->∞∞. In this limit we recover the elliptic PDE.. In this limit we recover the elliptic PDE. •In the absence of source terms, the temperature is bounded by initial and •In the absence of source terms, the temperature is bounded by initial and boundary conditions

boundary conditions

•Marching solutions are possible •Marching solutions are possible

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Hyperbolic PDEs

Consider the convection of a step Consider the convection of a step change in temperature:

change in temperature:

Initial and boundary conditions Initial and boundary conditions

Solution: Solution:

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Hy

(32)

Hy

Hype

perb

rbol

olic P

ic PDE

DEss (c

(con

ont’

t’d)

d)

•• UpstrUpstream eam condiconditions tions cancan

potentially affect the solution at a potentially affect the solution at a point x; downstream conditions do point x; downstream conditions do not

not

•• InleInlet condt conditionitions props propagate agate at aat a finite speed U

finite speed U

•Inlet condition is not felt at

•Inlet condition is not felt at locationlocation x until a time x/U

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Relation to Scalar Transport Equation

•• ContaContains all tins all three cahree canonicnonical PDE teal PDE termsrms

•• If Re is lIf Re is low and situaow and situation is stetion is steady, we get ady, we get an elliptican elliptic equation

equation

•• If diffusion If diffusion coefficient is coefficient is zero , we zero , we get a hyperbolget a hyperbolic equationic equation •• If Re is lIf Re is low and situaow and situation is unstion is unsteady, we get teady, we get a parabolica parabolic equation

equation

(34)

Components of CFD Solution



Geometry creation



Doma

Domain di

in discre

scretizatio

tization

n (mes

(mesh ge

h genera

neration)

tion)



Discr

Discretiza

etization

tion of

of gove

governing

rning equa

equations

tions



Solution of discrete equations; accounting for non-

Solution of discrete equations; accounting for

non-linea

linearities

rities and

and inter

inter-equa

-equation

tion coup

coupling

ling



(35)

Solution Process



Analytical solution gives us

φφ

(x,y,z,t). Numerical

solution gives us

φφ

only at

only at discrete

discrete grid points.

grid points.



The process of converting the governing partial

differential equation into discrete

differential equation into discrete algebraic equations

algebraic equations

is call

is call discretization.

discretization.



Discretization

Discretization involves

involves

»

» Di

Disc

scre

reti

tiza

zatio

tion

n of sp

of spac

ace us

e usin

ing me

g mesh ge

sh gene

nera

ratio

tion

n

»

» Dis

Discre

cretiz

tizati

ation

on of

of gov

govern

erning

ing eq

equat

uation

ions t

s to y

o yield

ield set

sets

s

of algebraic equations

(36)

Mesh Types

Regular and

body-fitted

meshes

Stair-stepped

representation of

complex geometry

(37)

Mesh types (cont’d)

Block-structured

structured

meshes

Unstructured

meshes

(38)

Mesh Types

Non-conformal

conformal

mesh

Hybrid mesh

Cell

shapes

(39)

Mesh Terminology

••

Node-based finite volume scheme

::

φφ

stored at vertex

(40)

Overview of Finite Difference Method



 Step Step 1: D1: Discreiscretizetize domadomain uin usingsing

a mesh. a mesh.

Unknowns are located at nodes Unknowns are located at nodes



 Step 2: ExpandStep 2: Expand

φφ

in Taylor seriesin Taylor series

about point 2 about point 2



 Subtracting equations yieldsSubtracting equations yields

Consider diffusion equation: Consider diffusion equation:

(41)

Finite Difference Method (cont’d)



 Step 3: Adding equations yieldsStep 3: Adding equations yields



 Drop truncated terms:Drop truncated terms:



 Step 4: Evaluate source term at point 2:Step 4: Evaluate source term at point 2:

Second order Second order truncation error truncation error

(42)

Finite Difference Method (cont’d)



 Step 5: Assemble discrete equationStep 5: Assemble discrete equation



 CommentsComments

»

» We cWe can wan writrite one one sue such eqch equatuation ion for efor each ach grigrid poid pointnt »

» BouBoundandary cry condonditiitions ons givgive us e us bouboundandary vry valualueses »

» SeSecocondnd-o-ordrder er acaccucuraratete »

(43)

Overview of Finite Volume Method

Consider the diffusion equation:

Step 1: Integrate over control

(44)

Finite Volume Scheme (cont’d)

Step 2: Make linear profile Step 2: Make linear profile

assumption between cell assumption between cell ce

centntroroididss foforr

φφ

. Assume S varies. Assume S varies linearly over CV

linearly over CV

Step 3: Collect terms and cast into Step 3: Collect terms and cast into

algebraic equation: algebraic equation:

(45)

Comments



Process starts with conservation statement over cell.

We find

φφ

such that it

such that it satisfies conservation. Thus,

satisfies conservation. Thus,

regardless of how coarse the mesh is, the finite

volume scheme always gives perfect conservation



This does not guarantee accuracy, however.



The pr

The proces

ocess of discr

s of discretiza

etization

tion yield

yields a flux bala

s a flux balance

nce

involving face values of the diffusion flux, for example:



Profile assumptions for

φφ

and S need not be the same.

e e e e x x

φ

∂∂

⎛

⎞⎞

−Γ

−Γ ⎜ ⎜ ⎟⎟

_∂∂

⎝

⎠⎠

(46)

Comments (cont’d)



As with finite difference method, we need to solve a

set of coupled algebraic equations



Though finite difference and finite

Though finite difference and finite volume schemes

volume schemes

use different procedures to obtain discrete equations,

we can use the same solution techniques to solve the

discrete equations

(47)

Closure

In this lecture we In this lecture we



 ConsiConsidered difdered different caferent canonicanonical PDEsl PDEs and examiand examined theirned their

behavior behavior



 Understood how these model equations relate to Understood how these model equations relate to our generalour general

scalar transport equations scalar transport equations



 Started an overview of the Started an overview of the important elements of any numericalimportant elements of any numerical

method method



 In the next lecture we will In the next lecture we will complete this overview and startcomplete this overview and start

looking more closely at the

looking more closely at the finite volume method for diffusionfinite volume method for diffusion problems.

(48)

Lecture 3: Overview of Numerical

Methods

(49)

Last time…



Examined important classes of partial differential

equations and understood their behavior



Saw how this knowledge would apply to the general

scalar transport equation



Started an overview of numerical

Started an overview of numerical methods including

methods including

mesh terminology and finite difference methods

(50)

This time…

We

We will

will continue

continue the

the overview

overview and

and examine

examine



Finite difference, finite volume and finite element

methods



Accuracy, consistency, stability and convergence of a

numerical scheme

(51)

Overview of Finite Element Method



 Consider diffusion equationConsider diffusion equation



 Let Let be be an an approximation approximation toto φφ 

 SSiincncee iis s aan n aapppprrooxxiimmaattioionn, , iit t ddoeoes s nnoot t ssaattiissffy y tthhe e ddiiffffuussiioonn

equation, and leaves a residual R: equation, and leaves a residual R:



 GalerGalerkinkin finitfinite element me element method miethod minimizenimizes R with res R with respect tspect to ao a

weight function: weight function: φ φ φ φ 2 2 2 2 d d S S RR dx dx

φ

Γ

+

+ =

=

(52)

Finite Element Method (cont’d)



 A family of weight functions WA family of weight functions W_ii, I = 1,…N, (N: number of grid, I = 1,…N, (N: number of grid

points) is used. This generates N discrete equations for the N points) is used. This generates N discrete equations for the N unknowns:

unknowns:

Weight function is local Weight function is local –

– i.i.e. zee. zero evro evererywywhehere re except close to i

except close to i

i+1 i+1 w w _i_i Element i Element i Element i-1 Element i-1 i i i-1 i-1

(53)

Finite Element Method (cont’d)



 In addition a local shape function NIn addition a local shape function N_ii is used is used to dto disciscretretizeize R.R.

Under a G

Under a Galerkialerkinn formuformulationlation, the weigh, the weight and shape fut and shape functionctions arens are chosen to be the same.

chosen to be the same.

N N _i_i

Shape function is non- Shape function is non- zero only in the vicinity zero only in the vicinity of node i => “local of node i => “local basis” basis” N N _i-1_i-1 i+1 i+1 i-1

i-1 _i_i

Element i Element i Element i-1

(54)

Finite Element Method (cont’d)



 The disThe discreticretizationzation procesprocess again les again leads to a seads to a set of alget of algebraicbraic

equations of the form: equations of the form:



 CommentsComments

»

» NotNote how e how the uthe use of se of a loca local baal basis rsis restestricricts thts the rele relatiationsonshiphip between a point i and its neighbors to only nearest neighbors between a point i and its neighbors to only nearest neighbors »

» AgAgaiain, wn, we hae have ve an aan alglgebebraraic eic equaquatition son set et to to sosolvlve –e – cacan un usese the same solvers as for finite

the same solvers as for finite volume and finite differencevolume and finite difference methods methods , , , , 1 1 1 1 , , 1 1 , , 11 i i i i i i i i i i i i i i i i i i i i ii

a

_φ

a

_φ

a

_φ

b

+ + ++ −− −−

=

+

(55)

Comparison of methods



 All three yield discrete algebraic equation sets which must beAll three yield discrete algebraic equation sets which must be

solved solved



 Local Local basis basis –– only only near-nenear-neighbor ighbor dependedependencence 

 Finite volume method is conservative; the others are notFinite volume method is conservative; the others are not 

 Order or accuracy of scheme depends onOrder or accuracy of scheme depends on

»

» TayTaylor slor serieries tes trunruncatcation iion in finn finite ite difdifferferencence sce schemhemeses »

» ProProfilfile e assassumpumptiotions ns in in finfinite ite voluvolume me schschemeemess »

(56)

Solution of Linear Equations



 Linear equation set has Linear equation set has two important characteristicstwo important characteristics

»

» MaMatrtrix iix is ss spaparsrse, e, mamay by be bae bandndeded »

» CoeCoeffifficiecients nts are are proprovisivisional onal for for nonnon-lin-linear ear probproblemlemss



 Two different approachesTwo different approaches

»

» DDiirreecct t mmeetthhooddss »

» IItteerraattiivve me meetthohoddss



 Approach defines “path to solution”Approach defines “path to solution”

»

(57)

Direct Methods



All d

All disc

iscret

retiza

izatio

tion

n sch

scheme

emes lea

s lead to

d to

Here

φ

is solution vector [

φ

₁₁

,,

φ

₂₂

,…,

φ

_N_N

]]

TT

..



Can invert

::



Inversion is O(N

33

) operation. Other more efficient

methods exist.

»

» Tak

Take adv

e advant

antage o

age of ban

f band str

d struct

ucture i

ure if it ex

f it exist

ists

s

»

(58)

Direct Methods (cont’d)



 Large storage and operation countLarge storage and operation count

»

» FoFor N r N grgrid id popoinintsts, m, musust st stotore re NxNxNN mamatrtrixix »

» OnlOnly sy stortore ne non-on-zerzero eo entrntries ies and and filfill pl pattatternern



 For non-linear problems,For non-linear problems, AA is provisional and is usually updatedis provisional and is usually updated

as a part of an outer loop as a part of an outer loop

»

» NoNot wot wortrth soh solvlving ing sysyststem tem too “oo “exexacactltly”y”



(59)

Iterative Methods



 Guess and correct philosophyGuess and correct philosophy 

 Gauss-Seidel scheme is typical:Gauss-Seidel scheme is typical:

»

» ViVisisit t eaeach ch grgrid id popoinintt Update using

Update using »

» SweSweep reep repeapeatedtedly tly throuhrough grgh grid poid pointints uns until ctil convonvergeergencence criterion is met

criterion is met »

» In eaIn each swch sweepeep, poi, points ants alrelready vady visitisited haed have neve new vaw valueslues; poi; pointsnts not yet visited have old values

(60)

Iterative Methods (cont’d)



 JacobiJacobi scheme scheme is similis similar to Gaar to Gauss-Suss-Seidel sceidel scheme but dheme but does notoes not

use latest available values use latest available values

»

» All All valvalues ues are are updaupdated ted simsimultultaneoaneouslusly at y at end end of sof sweeweep.p.



 Iterative are not guaranteed to converge to a solution unlessIterative are not guaranteed to converge to a solution unless

Scarborough criterion

(61)

Scarborough Criterion



 Scarborough criterion states that convergence of an iterativeScarborough criterion states that convergence of an iterative

scheme is guaranteed if: scheme is guaranteed if:



(62)

Gauss-Seidel Scheme



 No need to store coefficient matrixNo need to store coefficient matrix 

 Operation count per sweep scales Operation count per sweep scales as O(N)as O(N) 

 However, convergence, even when guaranteed, is slow for largeHowever, convergence, even when guaranteed, is slow for large

meshes meshes



(63)

Accuracy



 While looking at finite difference methods, we wrote:While looking at finite difference methods, we wrote:



 Halving grid size reduces error Halving grid size reduces error by factor of four for second-orderby factor of four for second-order

scheme scheme



 CannoCannot say what at say what absolutbsolute error is –e error is – truncatruncation errtion error only giveor only givess

rate of decrease rate of decrease Second- Second- order order truncation truncation error error

(64)

Accuracy



 OrdOrder of discer of discretretizaizatiotionn schscheme iseme is n n if if truncation truncation error error is is O(O(∆∆xxnn )) 

 When more than one term is involved, the order of theWhen more than one term is involved, the order of the

discr

discretizatetizationion schemscheme is e is that that of tof thehe lowest order lowest order term.term.



 AccurAccuracy is a properacy is a property of the discrty of the discretizaetizationtion schemscheme, not the pathe, not the path

to solution to solution

(65)

Consistency



 A A discrediscretizatitizationon schemscheme ie is cs consistonsistent ent if if the the trunctruncation ation errorerror

vanishes as

vanishes as ∆∆x ->0x ->0



 Does not always happen: What Does not always happen: What if truncation error is O(if truncation error is O(∆∆x/ x/ ∆∆t) ?t) ? 

 ConsiConsistencstency is a property of the disy is a property of the discreticretizationzation schemscheme, not thee, not the

path to solution path to solution

(66)

Convergence



 Two uses of the termTwo uses of the term

»

» ConConververgengence tce to a meo a mesh-sh-indeindependpendent ent solsolutiution ton throhrough mugh meshesh refinement

refinement »

» ConConververgengence of ce of an itan iteraerativtive sce schemheme to e to a fia final unal unchnchanganginging answer (or one meeting convergence criterion)

answer (or one meeting convergence criterion)



(67)

Stability



 Property of the path to solutionProperty of the path to solution 

 Typically used to characterize iterative schemesTypically used to characterize iterative schemes 

 Depending on the characteristics of Depending on the characteristics of the coefficient matrix, errorsthe coefficient matrix, errors

may either be damped or may grow during iteration may either be damped or may grow during iteration



 An iterative An iterative scheme scheme is unstable is unstable if it if it fails to fails to produce a produce a solution tosolution to

the discrete equation set the discrete equation set

(68)

Stability



Also possible to speak of the

Also possible to speak of the stability of unsteady

stability of unsteady

schemes

»

» Unstable

Unstable : when solving a time-dependent problem,

: when solving a time-dependent problem,

the solution “blows up”



Von-Neumann (and other) stability analyses determine

whether linear systems stable under various

iteration/time-stepping schemes



For non-linear/coupled problems, stability analysis is

difficult and not much used

»

» Tak

Take guid

e guidanc

ance from l

e from line

inear ana

ar analys

lysis in ap

is in appro

propri

priate

ate

parameter range; intuition

(69)

Closure



 This time we completed an overview of the numericalThis time we completed an overview of the numerical

dis

discrecretiztizatiationon and soluand solutiotion procesn processs »

» DDomomaiain din discscreretitizazatitionon »

» DiDiscscreretitizazatitionon of of gogovevernrnining g eqequauatitiononss »

» SoSolulutition oon of lif linenear aar alglgebebraraic sic setet »

» PrPropeopertrtieies os of f didiscscreretitizazatitionon anand pd patath th to so soluolutitionon

–

– AccuAccuracy, racy, consconsistenistency, cy, conveconvergencrgence, se, stabiltabilityity 

 Next timNext time, we will stare, we will start looking at finit looking at finite volume diste volume discreticretizationzation ofof

diffusion equation diffusion equation

(70)

Lect

Lecture

ure 4: T

4: The D

he Diffu

iffusion

sion Equ

Equati

ation –

on – A

A

First Look

(71)

Last Time…



We completed an overview of the numerical

dis

discre

cretiz

tizati

ation

on and so

and solut

lution pr

ion proce

ocess

ss

»

» Do

Doma

main d

in dis

iscr

cret

etiz

izat

atio

ion

n

»

» Di

Disc

scre

reti

tiza

zati

tion

on of

of go

gove

vern

rnin

ing

g eq

equa

uati

tion

ons

s –

– fi

fini

nite

te

difference, finite volume, finite element

»

» So

Solu

luti

tion o

on of li

f line

near a

ar alg

lgeb

ebra

raic s

ic set

et

»

» Pr

Prop

oper

erti

ties

es of d

of dis

iscr

cret

etiz

izat

atio

ion

n an

and p

d pat

ath to

h to so

solu

luti

tion

on

–

(72)

This Time…

We will



Apply the finite volume scheme to the

Apply the finite volume scheme to the steady diffusion

steady diffusion

equation on Cartesian structured meshes



Examine the properties of the

Examine the properties of the resulting discretization

resulting discretization



(73)

2D Steady Diffusion

•• Con

Consid

sider st

er stead

eady dif

y diffus

fusion w

ion with a

ith a

source term:

••

Here

•• Int

Integr

egrate o

ate over

ver con

contro

trol vo

l volum

lume to

e to

yield

(74)

2D Steady Diffusion



(75)

Discrete Flux Balance

••

Writing integral over control volume:

•Compactly:

(76)

Discrete Flux Balance (cont’d)



Area vectors given by:



(77)

Discretization



 AssumeAssume φφ varies linearlyvaries linearly

between cell centroids between cell centroids



 Note:Note:

»

» SySymmmemettry ry of of ((P, P, E ) E ) anandd (P,W) in flux expression (P,W) in flux expression »

» OOpppposositite se sigigns ns on on ((P,P,EE)) and (P,W) terms

(78)

Source Linearization



Source term must be

liline

near

ariize

zed

d as

as::



Assume S

_P_P

<0



(79)

Final Discrete Equation

P P N N S S E E W W

(80)

Comments



Discrete equation reflects balance of flux*area with

generation inside control volume



As in 1-D case, we need fluxes at cell faces



These

These are wr

are written

itten in term

in terms of ce

s of cell-ce

ll-centroid

ntroid value

values usin

s using

g

profile assumptions.

(81)

Comments (cont’d)



Formulation is conservative: Discrete equation was

derived by enforcing conservation. Fluxes balance

source term regardless of mesh density



For a structured mesh, each point P is coupled to its

four nearest neighbors. Corner points do not enter the

formulation.

(82)

Properties of Discretization



a

_P_P

, a

_nb_nb

have same sign: This implies that if neighbor

have same sign: This implies that if neighbor φ

φ

goes up,

goes up, φ

φ

_P_P

also goes up



If S=0:



Thus

Thus φ

φ

is bounded by neighbor values, in keeping with

properties of elliptic partial differential equations

(83)

Pr

Prope

operti

rties o

es of Di

f Discr

screti

etizat

zation

ion (co

(cont’

nt’d)

d)



What about

What about Scarborough Criterion ?

Scarborough Criterion ?

Satisfied in

the equality

What about

this?

(84)

Boundary Conditions

Flux Balance Flux Balance

Different boundary conditions Different boundary conditions require different representations require different representations of

of JJ

b b

(85)

D

Diirriicch

hlleett B

BC

Css



 DirDirichichletlet bounboundardary condy conditiition:on:

φ

φ_b_b == φφ_given_given



 Put in the requisite flux into thePut in the requisite flux into the

near-boundary cell balance near-boundary cell balance

(86)

Di

Diri

rich

chle

lett BC

BC’s (c

’s (con

ont’

t’d)

d)

P P nbnb nb nb a a

_>

∑

aa For near-boundary For near-boundary cells: cells: Satisfies Scarborough Satisfies Scarborough Criterion ! Criterion ! Also,

Also, φφ_P_P bounded bybounded by interior neighbors and interior neighbors and boundary value in the boundary value in the absence of source terms absence of source terms

(87)

Neumann BC’s



Neumann boundary

conditions : q

_b_b

given



Replace J

_b_b

in cell

balance with given flux

(88)

Neumann BC’s (cont’d)

P P nbnb nb nb a a

₌

∑

aa

For Neumann boundaries For Neumann boundaries

So inequality constraint in So inequality constraint in Scarborough criterion is not Scarborough criterion is not satisfied

satisfied Also,

Also, φφ_P_Pis not boundedis not bounded by interior neighbors and by interior neighbors and boundary value even in boundary value even in the absence of source the absence of source te

termrms s –– ththis is is is isis ffininee because of the added because of the added flux at the boundary flux at the boundary