PN 5013 Aerodinamika Numerik-Lecture8 Linear Hyperbolic
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Transcript of PN 5013 Aerodinamika Numerik-Lecture8 Linear Hyperbolic
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Page 1 Moelyadi - 09.11.2013
CFD LECTURE AE 4012 NUMERICAL AERODYNAMICS
Lecture 8 : Hyperbolic Numerical Schemes
Farfield boundary
Body/ Solid boundary
Grids /
Mesh
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Page 2 Moelyadi - 09.11.2013
Computational Modelling
Real Words Physics Numerical Simulation
Flow Models
Dynamic
approximation
Spatial
approximation
Steadiness
approximation
Space
discretization Mesh definition
Equation
discretization Definition of
Numerical schemes
Mathematical
Model
Discretization
Techniques
Resolution of
discrete system
of Equations.
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Page 3 Moelyadi - 09.11.2013
Review of Mathematical Models
Linear equations Non-Linear equation System equations
1. Linear convection
3. Transport (unsteady
convection-diffusion)
2. Linear diffusion
(heat conduction)
4. Laplace
5. Wave
1. Inviscid Burgers
2. Burgers
1. Unsteady Inviscid
compressible flow
Where p is pressure and E
is total energy per unit
volume given by
and g is ratio of specific
heats
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Page 4 Moelyadi - 09.11.2013
Road Map
Implicit
ADI Beam Warming
Hyperbolic
Linear problems Non-linear problems
Explicit
Eulers FTFS Eulers FTCS Upwind Lax Method Midpoint Leapfrog Lax-Wendroff
Explicit
Lax Method Lax-Wendroff MacCormack
Implicit
Eulers FTCS Crank Nicolson
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Page 5 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
Linear convection
Eulers Forward Time Forward Space (FTFS)
x
TTu
t
TTn
i
n
i
n
i
n
i
1
1
forward time forward space
nininini TTx
tuTT
1
1
Stability factor
Based on Von Neumann stability analysis, this method is
unconditionally unstable
Explicit Formulation
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Page 6 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
Linear convection
Eulers Forward Time Central Space (FTCS)
x
TTu
t
TTn
i
n
i
n
i
n
i
11
1
forward time forward space
nininini TTx
tuTT 11
1
2
Stability factor
Based on Von Neumann stability analysis, this method is
unconditionally unstable
Explicit Formulation
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Page 7 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
Linear convection
Eulers Forward Time Backward Space (FTCS)/ Upwind Method
x
TTu
t
TTn
i
n
i
n
i
n
i
1
1
forward time forward space
nininini TTx
tuTT 1
1
Stability factor
Based on Von Neumann stability analysis, this method is
conditionally stable
numberCourantccx
tu
1
Explicit Formulation
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Page 8 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
Linear convection
Eulers FTCS nininini TTx
tuTT 11
1
2
unconditionally unstable
Lax Method
ninin
i
n
in
i TTx
tu
TTT 11
111
22
Lax method
conditionally stable
numberCourantccx
tu
1Stable
Taking an average value for of the Eulers FTCS method n
iT
Explicit Formulation
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Page 9 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
numberCourantccx
tu
1
Linear convection
nininini TTx
tuTT 11
11
Midpoint Leapfrog Method (CTCS)
Stable
x
TTu
t
TTn
i
n
i
n
i
n
i
22
11
11
conditionally stable
Difficulty in starting procedure require a large computer storage
Explicit Formulation
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Page 10 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
numberCourantccx
tu
1
Linear convection
x
Tu
t
T
Lax-Wendroff method
Stable
32
2
2
)(!2
)(),(),( tO
t
t
Tt
t
TtxTttxT
2
22
2
2
x
Tu
t
T
xu
x
T
tu
t
T
2
22
21
2
)(
x
Tu
tt
x
TuTT
n
i
n
i
First derivative
Second derivative
Central difference
2
1122111
)(
2)(
2
1
2 x
TTTtu
x
TTtuTT
n
i
n
i
n
i
n
i
n
in
i
n
i
Explicit Formulation
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Page 11 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
Linear convection
Eulers FTCS method
Implicit Formulation
x
TTu
t
TTn
i
n
i
n
i
n
i
1
1
1
1
1
n
i
n
i
n
i
n
i TcTTcT
1
1
11
12
1
2
1
Unconditionally stable
numberCourantx
tuc
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Page 12 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
Linear convection
Crank-Nicolson method
Implicit Formulation
x
TT
x
TTu
t
TTn
i
n
i
n
i
n
i
n
i
n
i
222
1 111
1
1
1
1
n
i
n
i
n
i
n
i
n
i
n
i cTTcTcTTcT 111
1
11
14
1
4
1
4
1
4
1
Unconditionally stable
numberCourantx
tuc
])(,)[( 22 xt
accuracy
n
i
n
i
n
i
n
i
x
T
x
Tu
t
TT1
1
2
1
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Page 13 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
Linear convection
ADI method
Splitting Method
n
i
n
i
n
i
n
i
n
i
n
i cTTcTcTTcT 11114
1
4
1
4
1
4
12
12
12
1
Unconditionally stable
numberCourantx
tuc
21
21
21
11
1
1
11
14
1
4
1
4
1
4
1
n
i
n
i
n
i
n
i
n
i
n
i cTTcTcTTcT
])(,)[( 22 xt
accuracy
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Page 14 Moelyadi - 09.11.2013
Linear Hyperbolic Equation
x
TTu
TTT ni
n
i
t
n
i
n
i
n
i
2
)( 11
2
11212
1
Linear convection
Richtmyer/ Lax-Wendroff method
Multistep Method
ninininini TTcTTT 11114
1
2
12
1
conditionally stable
1
x
tuc
2121 1112
1
n
i
n
i
n
i
n
i TTcTT
Used for non-linear problems
Richtmyer formulation
x
TTu
t
TTn
i
n
i
n
i
n
i
2
21
21
11
1
Lax-Wendroff formulation
ninininini TTcTTT 11112
1
2
12
1
21
212
12
1
21
2
11
n
i
n
i
n
i
n
i TTcTT
])(,)[( 22 xt
accuracy