Page 1 Moelyadi - 09.11.2013

CFD LECTURE AE 4012 NUMERICAL AERODYNAMICS

Lecture 8 : Hyperbolic Numerical Schemes

Farfield boundary

Body/ Solid boundary

Grids /

Mesh

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.

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

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

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

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

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

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

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

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

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

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

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

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