Post on 22-Oct-2014
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