Chap 10

Chapter 10Chapter 10

Nonlinear Nonlinear Convection-Dominated Convection-Dominated

ProblemsProblems

10.1 Burgers’ Equation10.1 Burgers’ Equation One-dimensional Burgers’ equation

Conservative form

0x

u

x

uu

t

u2

2

22

2

2

22

u0.5F ;0x

u

x

F

t

u

0x

u

2

u

xt

u

Inviscid Burgers’ EquationInviscid Burgers’ Equation One-dimensional inviscid Burgers’ equation

Larger values of convect faster and overtake slower

Multi-valued solution may occur Postulate a shock to allow the development of

discontinuous solutions

0x

uu

t

u

uu

Inviscid Burgers’ EquationInviscid Burgers’ Equation Formation of multi-valued solution

The nonlinearity allows discontinuous solutions to develop

Shock-fitting

a

b

shock

t = t0t = t1 t = t2

Viscous Burgers’ EquationViscous Burgers’ Equation Viscous term reduces the amplitude in

high gradient regions Prevents multi-valued solutions from

developing (second derivative increases faster than first derivative)

t = t0t = t1 t = t2

10.1.2 Explicit Schemes10.1.2 Explicit Schemes FTCS scheme (non-conservative)

FTCS (conservative form)

0x2

uu2u

x2

uuu

t

uu n1j

nj

n1j

n1j

n1j

nj

nj

1nj

)()(

2

n1j

nj

n1j

n1j

n1j

nj

1nj

u50F

0x2

uu2u

x2

FF

t

uu

.

)(

Explicit SchemesExplicit Schemes Four-point Upwind Scheme

Truncation errors O(x2) if q 0.5 O(x3) if q = 0.5

x3

FF3F3Fq

x2

FFFL 0u

x3

FF3F3Fq

x2

FFFL 0u

2j1jj1j1j1j(4)x

1jj1j2j1j1j(4)x

)(,

)(,

Lax-Wendroff SchemeLax-Wendroff Scheme Inviscid Burgers’ equation for unsteady one-

dimensional shock flows

Replace temporal derivative by equivalent spatial derivative (more complicated for nonlinear case)

2u0.5F ;0x

F

t

u

u

FA

x

FA

xt

F

xt

u

x

FA

x

F

u

F

t

u

u

F

t

F

t

F

xx

F

tt

u

2

2

2

2

;

Chain rule

Lax-Wendroff SchemeLax-Wendroff Scheme Central-difference discretization

For Burgers’ equation

2

n1j

nj21j

nj

n1j21j

21j21j

1jj

n1j

nj

21jj1j

nj

n1j

21j

21j21j

21j

21j

21j

21j

x

FFAFFA

xx

xx

FFA

xx

FFA

xx

x

FA

x

FA

x

FA

x

)()( //

//

//

//

/

/

/

/

uu50A & uu50A uA 1jj1/2j1jj1/2j )(.)(.

Lax-Wendroff SchemeLax-Wendroff Scheme Temporal derivative

Inviscid Burgers’ equation

Rearrange

2

n1j

nj21j

nj

n1j21j

nj

1nj

nj

1nj

2

2nj

1nj

x

FFAFFA

2

t

t

uu

x

FA

x2

t

t

uu

t

u

2

t

t

uu

t

u

)()(

)(

//

0x2

FF

x

FFAFFA

2

t

t

uu

x

F

t

un

1jn

1j

2

n1j

nj21j

nj

n1j21j

nj

1nj

)()( //

)()(.)(. //n

1jnj21j

nj

n1j21j

2n

1jn

1jnj

1nj FFAFFA

x

t50FF

x

t50uu

Lax-Wendroff SchemeLax-Wendroff Scheme Linear pure convection equation

Nonlinear - inviscid Burgers’ equation

Equivalent two-stage algorithm (more economical)

)()(.)(. //n

1jnj21j

nj

n1j21j

2n

1jn

1jnj

1nj FFAFFA

x

t50FF

x

t50uu

)(.)(.

)(.)(.

n1j

nj

n1j

2n1j

n1j

nj

1nj

n1j

nj

n1j

2n

1jn

1jnj

1nj

TT2TC50TTC50TT

TT2Tx

tu50TT

x

tu50TT

)(

)(.)(.

*/

*/

*/

21j21jnj

1nj

nj

n1j

n1j

nj21j

FFx

tuu

FFx

t50uu50u

Burgers’ EquationBurgers’ Equation Thommen’s extension of Lax-Wendroff

scheme for viscous flow problems

Error in textbook Stability limit

2

n1j

nj

n1j21j21j

nj

1nj

n2j

n1j

nj

n1j

nj

n1j

nj

n1j

n1j

nj21j

x

tswhere

uu2usFFx

tuu

uu2u50uu2u50s50

FFx

t50uu50u

)()(

)](.)(.[.

)(.)(.

*/

*/

*/

)/()( xA2xt or x2tAt 222

10.1.3 Implicit Schemes10.1.3 Implicit Schemes Burgers’ equation (viscous) Crank-Nicolson implicit formulation

Thomas algorithm for tridiagonal matrices cannot be used directly due to the appearance of nonlinear implicit term

Use Taylor series expansion of at nth time- level to convert to tridiagonal form

2xxx

nj

1nj

1nj

1nj

njxx

1nj

njx

1nj

x1 2 1L x21 0 1L uuu

uuL50FFL50t

u

/),,(),/(),,(,

)(.)(.

1njF

1njF

Crank-Nicolson SchemeCrank-Nicolson Scheme Taylor-series expansion (linearlization of F)

Linear tridiagonal system (in terms of u or u)

nj

n

j

21nj

nj

1nj

n

j

2

22

n

j

nj

1nj

uu

FA tOuAFF

or t

Ft50

t

FtFF

),(

.

njxx

nj

1njxx

1nj

njx

1nj

1nj

njxx

1nj

nj

njx

1nj

uL50uuLuuLt50u

or uuL50uuF2L50t

u

.)(.

)(.)(.

Crank-Nicolson SchemeCrank-Nicolson Scheme Thomas algorithm

The matrix coefficients must be reevaluated at every time step (to recover nonlinearity of the equation)

Truncation error O(t2, x2)

Unconditionally stable in Von Neumann sense (linear)

n1j

nj

n1j

nj

n1j

nj

nj

n1j

nj

nj

1n1j

nj

1nj

nj

1n1j

nj

su50us1su50d

s50ux

t250c

s1b

s50ux

t250a

ducubua

.)(.

..

..

Generalized Crank-NicolsonGeneralized Crank-Nicolson Mass operator and four-point upwind

Truncation error O(t2, x2)

2

n1j

nj

n1jn

jxx

1jj1j2j1j1j(4)x

x

1nj

njxx

1nj

nj

nj

4x

1nj

x

x

uu2uuL

x3

FF3F3Fq

x2

FFFL 0u

21 M

uuL50uuF2L50t

uM

)(,

),,,(

)(.)(. )(

Generalized Crank-NicolsonGeneralized Crank-Nicolson Quadridiagonal system of equations – can be

solved using generalized Thomas algorithm

n1j

nj

n1j

nj

n1j

nj

nj

nj

n1j

nj

n2j

nj

nj

1n1j

nj

1nj

nj

1n1j

nj

1n2j

nj

us50us21us50d

s50ux

t

6

q250c

s2ux

tq501b

s50ux

tq50250a

ux

t

6

qe

ducubuaue

).()().(

.).(

.

.)..(

Artificial DissipationArtificial Dissipation Crank-Nicolson with additional dissipation

For small values of viscosity (high-Re), it is desirable to add some artificial dissipation

Modified Crank-Nicolson

Choose a empirically

)(. 1nj

njxxa FFtL50

njxx

njx

1nj

njxxa

1njxx

1nj

nj

4x

1njx

utL50uM

uutLuLuuLt50uM

.

)()(. )(

10.1.4 BURG: 10.1.4 BURG: Numerical ComparisonNumerical Comparison

Propagation of a shock wave governed by viscous Burgers’ equation

Exact solution

0txu 01txu s B.C.

xx0 ,0

0xx , 10xuxu

max0

),(,.),(

),()(

maxmax

max

1

t

x50dutxG

de

det

x

u

0

2

0

G50

G50

Re;)(.

)(),;(

Re.

Re.

Burgers’ EquationBurgers’ Equation

BURG: Numerical ComparisonBURG: Numerical Comparison ME = 1, FTCS scheme ME = 2, two-stage Lax-Wendroff scheme ME = 3, Explicit four-point upwind scheme ME = 4, Crank-Nicolson (CN-FDM): = 0, q = 0 ME = 4, Crank-Nicolson (CN-FEM): = 1/6, q = 0 ME = 4, Crank-Nicolson, Mass Operator (CN-MO): = 1/12, q =

0 ME = 4, Crank-Nicolson, 4-pt. Upwind (CN-4PU): = 0, q = 0.5 ME = 5, Crank-Nicolson plus additional dissipation

Note: Optimum and q (locally freezing nonlinear coefficients)

2

jopt

2

jopt

x

tu

4

1

2

1q

x

tu

12

1

6

1

Burgers’ EquationBurgers’ Equation

Burgers’ Equation Burgers’ Equation Propagating Shock SolutionPropagating Shock Solution

Rcell = 1.0, C = 0.25

Burgers’ Equation: Propagating ShockBurgers’ Equation: Propagating Shock

Rcell = 100, C = 1.0

Rcell = 3.33, C = 1.0

Velocity distribution at t = 2.0; RVelocity distribution at t = 2.0; Rcellcell = 100 = 100

10.2 Systems of Equations10.2 Systems of Equations

Continuity equation Momentum equations Energy equation Equation of state (compressible flows) Turbulent kinetic energy equation Rate of turbulent energy dissipation equation Reynolds stresses equations Multiphase flows Chemical reactions

Systems of EquationsSystems of Equations 1D unsteady compressible inviscid flow Continuity equation, x-momentum equation,

energy equation

v

p

2

2

2

C

C ;

uu501

p

pu

u

F ;

u501

p

uq

0x

F

t

q

).(.

)(

Two-Stage Lax-WendroffTwo-Stage Lax-Wendroff Single equation

System of equations

)(

)(.)(.

*/

*/

*/

21j21jnj

1nj

nj

n1j

n1j

nj21j

FFx

tqq

FFx

t50qq50q

)(

)(.)(.

*/

*/

*/

21j21jnj

1nj

nj

n1j

n1j

nj21j

FFx

tuu

FFx

t50uu50u

Lax-Wendroff Scheme with Lax-Wendroff Scheme with Artificial ViscosityArtificial Viscosity

Continuity equation

X-momentum equation

Energy equation

Crank-Nicolson SchemeCrank-Nicolson Scheme System of equations

Linearization

33 block tridiagonal system (solved by block Thomas algorithm)

)()(. 1n1j

1n1j

n1j

n1j

nj

1nj FFFF

x

t250qq

q

FA

qAFF 1nn1n

(33 matrix)

1nj

nj

1nj

n1j

n1j

1n1j1j

1nj

1n1j1j

qqq

FFx

t50qA

x

t250qIqA

x

t250

)(...

Crank-Nicolson SchemeCrank-Nicolson Scheme Use Von Neumann analysis for the

linearized equation

Amplification matrix

Numerical Stability

q

FA ;0

x

qA

t

q

sin.

sin.

Ax

ti501

Ax

ti501

G

m all for 01 satisfy G of igenvaluesE m .

10.3 Group Finite Element Method10.3 Group Finite Element Method

Conventional finite element method introduces a separation approximate solution (trial function, interpolation function) for each dependent variable

Galerkin method produces large numbers of products of nodal values of dependent variables, particularly from the nonlinear convective terms

Inefficient, time-consuming Group finite element formulation is effective

in dealing with convective nonlinearities

Group Finite Element MethodGroup Finite Element Method Group finite element formulation

1. The equations are cast in conservative form

2. A single approximation solution is used for the group of terms in the differential terms (i.e., approximate F directly instead of the nonlinear convective term uu/x)

One-dimensional Group Formulation

FF & uul

lll

ll

Group Finite Element MethodGroup Finite Element Method One-dimensional Group Formulation

Conventional finite element

)(

)(.

21j

21j

1j1j1j1jjx

jxxjxj

x

uux4

1

x2

uuuu50FL

0uLFLdt

duM

x2

uu

3

uuu

x

uu 1j1j1jj1j )(

Conservative form

Non-conservative form

One-dimensional Burgers’ equationOne-dimensional Burgers’ equation

Conventional and group FEMs

10.4 2D Burgers’ Equation10.4 2D Burgers’ Equation

Two-dimensional Burgers’ equation

Equivalent to 2D momentum equations for incompressible laminar flow with zero pressure gradient

2

2

2

2

2

2

2

2

y

v

x

v

y

vv

x

vu

t

v

y

u

x

u

y

uv

x

uu

t

u

2D Burgers’ Equation2D Burgers’ Equation Exact solution Use Cole-Hopf transformation

Transform the 2D Burgers’ equation into one single equation – 2D diffusion equation

y

2v , x

2u

0yxt 2

2

2

2

2D Burgers’ Equation2D Burgers’ Equation Steady 2D Burgers’ equation

Exact solution

0yx 2

2

2

2

)cos(][

)sin(][

)cos(][

)cos(][

)cos(][

)()(

)()(

)()(

)()(

)()(

yeeaxyayaxaa

yeeaxaa 2v

yeeaxyayaxaa

yeeayaa 2u

yeeaxyayaxaa

00

00

00

00

00

xxxx54321

xxxx543

xxxx54321

xxxx542

xxxx54321

Exact solution for 2D Burgers’ equationExact solution for 2D Burgers’ equation

2D Burgers’ Equation – Exact u2D Burgers’ Equation – Exact ua1= a2= 1.3*1013, a3= a4= 0, a5 = 1, = 25, x0 =1, = 0.04

2D Burgers’ Equation – Exact v2D Burgers’ Equation – Exact va1= a2= 1.3*1013, a3= a4= 0, a5 = 1, = 25, x0 =1, = 0.04

Multidimensional Group FEMMultidimensional Group FEM Two-dimensional Burgers’ equation

Approximate solutions for (u,v), and groups (u2, uv, v2) and the components of S

For example (bilinear for rectangular elements)

)(.,

)(.

},{},,{},,{2222

22

2

2

2

2

vuv50vuu50S

vuvG uvuF vuq

0Sy

q

x

q

y

G

x

F

t

q

4

1llluvuv ),()(

Galerkin Finite ElementGalerkin Finite Element Linear (Chapter 9)

Nonlinear (Group FE formulation)

The equations are treated as linear at the level at which the discretization take place (but indeterminate)

Substitution for the nodal groups in terms of the unknown nodal variables introduces the nonlinearity but also makes the system determinate

SMMqLMLMGLMFLMRHS

RHSt

qMM

yxyyxxxyyxxy

kjyx

)(

,

kjyyxyxxyxyxxykj

yx TLMLMLvMLuMt

TMM ,

,

)[

Split SchemesSplit Schemes Two-dimensional Burgers’ equations Similar to those used in Chapters 8 and 9 Additional complication due to nonlinearity Generalized FEM/FEM with mass

operators Mx and My

Tyyyy

xxxx

21 M

21 M

),,(

),,(

Pseudo-Transient FormulationPseudo-Transient Formulation

Use pseudo-transient formulation (sect 6.4) for steady-state solution

For steady-state problems, unsteady formulation provides an equivalent underrelaxation parameter for steady iterative schemes

For steady-state solutions, it is desirable to use a simple time discretization (such as two-level fully implicit scheme with = 1) to simplify the formulation

Pseudo-Transient FormulationPseudo-Transient Formulation Two-level fully implicit scheme ( = 1)

Linearize the nonlinear terms F, G, and S in (RHS)n+1

1n1n

kjyx RHS

t

qMM

,

qCSS

qBGG

qAFF

1nn1n

1nn1n

1nn1n

Pseudo-Transient FormulationPseudo-Transient Formulation Linearization (Jacobian matrices A, B, C)

Approximate Factorization

)(

)(.

)(.,

)(.

},,{},,{},,{

22

22

2222

22

v3uuv2

uv2vu350

q

SC

v20

uv

q

GB ,

uv

0u2

q

FA

vuv50vuu50S

vuvG uvuF vuq

*,,

*,

)].([

)()].([

kj1n

kjyyyyy

nkjxxxxx

qqCM50LBLtM

RHStqCM50LALtM

Pseudo-Transient FormulationPseudo-Transient Formulation Further simplification to reduce CPU time

Use the same left-hand-side for each scalar component

Perform only one factorization (BANFAC) for different components

Does not affect the steady-state solution since (RHS)n = 0 in the steady state limit

10

01vuC

, 10

01vB ,

10

01uA

22

TWBURG: Numerical SolutionTWBURG: Numerical Solution Two-dimensional Burgers’ equations Steady state solution with the following split algorithm

Solution domain

1 x 1 , 0 y ymax , ymax= /6

Use exact solution for the boundary conditions Initial conditions obtained from linear interpolation of

the boundary condition in the x-direction

*,,

*,

)].([

)()].([

kj1n

kjyyyyy

nkjxxxxx

qqCM50LBLtM

RHStqCM50LALtM

Computer Program - TWBURGComputer Program - TWBURG

Approximate FactorizationApproximate Factorization

Error Distributions at y/yError Distributions at y/ymaxmax = 0.4 = 0.4