Flux Form Finite Difference / Flux Limiters

EP711 Supplementary MaterialTuesday, February 28, 2012

Jonathan B. Snively Embry-Riddle Aeronautical University

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Contents

• Flux Form Finite Difference Methods• Flux Limiters

EP711 Supplementary MaterialTuesday, February 28, 2012

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“Flux Form” (for Conservative Advection) For a scalar quantity (such as density), the “flux” f depends on q(x,t). The continuity equation can be written as:

Perhaps the best-known conservation law is the continuity equation for neutral mass density:

@⇢

@t= �r · (⇢~v)

@⇢

@t

= � @

@x

(⇢v) = �v

@⇢

@x

� ⇢

@v

@x

In 1-Dimension:

@q

@t= �r · ~f

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Conservation Laws and Continuity Equations

Qfx fx + (∂fx/∂x) dx

dx

In 1D

@q

@t= �r · ~f

@q

@t

= �@f(q)@x

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Finite Volume Methods

QiFi-1/2 Fi+1/2

xi xi+1/2xi-1/2

Qi is a cell average of quantity q at a given time step tn:

Fi+1/2 is the flux through the lateral cell interface at tn.

Q

n

i

' 1�x

Zxi+1/2

xi�1/2

q(x, t

n

)dx

d

dt

Zxi+1/2

xi�1/2

q(x, t)dx = f(q(xi�1/2, t))� f(q(x

i+1/2, t))

tn

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Finite Volume Methods

QiFi-1/2 Fi+1/2

xi xi+1/2xi-1/2

Qi is a cell average of quantity q at a given time step tn:

Fi+1/2 is the flux through the lateral cell interface at tn.

Q

n

i

' 1�x

Zxi+1/2

xi�1/2

q(x, t

n

)dx

tn

Q

n+1i = Q

ni �

�t

�x

(Fni+1/2 � F

ni�1/2)

Finite Volume Methods are based on difference approximations of this form.

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Finite Volume MethodsRecall Qi is a cell average of quantity q at a time step tn:

Q

n

i

' 1�x

Zxi+1/2

xi�1/2

q(x, t

n

)dx

Q

n+1i = Q

ni �

�t

�x

(Fni+1/2 � F

ni�1/2)

Finite volume methods seek to approximate “Fluxes” F, to obtain an average of fluxes at the cell interfaces occurring over a single time step:

F

ni�1/2 '

1�t

Z tn+1

tn

f(q(xi�1/2, t))dt

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Flux Form Finite DifferencingFinite difference methods can also be expressed in the same flux form:

Q

n+1i = Q

ni �

�t

�x

(Fni+1/2 � F

ni�1/2)

In this case, the fluxes F can be derived directly from difference method solutions (i.e., for Lax, Lax-Wendroff, Upwind, etc.).

The many benefits of this approach will be seen shortly!

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Upwind Method for Advection

The flux form of the Upwind Method can be defined:

The Upwind solution for the constant coefficient advection equation (Conservation Law) can be written as:

Q

n+1i = Q

ni �

�t

�x

(Fni+1/2 � F

ni�1/2)

Q

n+1i = Q

ni �

u�t

�x

(Qni �Q

ni�1)

Fni�1/2 = uQn

i�1 That was easy...

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Lax-Friedrichs (Flux Form)

A flux form of the Lax-Friedrichs Method can be defined:

F

ni�1/2 =

12[f(Qn

i�1 + f(Qni )]� �x

2�t

(Qni �Q

ni�1)

Spatial Average Diffusive Flux

Q

n+1i =

12(Qn

i�1 + Q

ni+1)�

�t

2�x

[f(Qni+1)� f(Qn

i�1)]

The Lax-Friedrichs solution for the continuity equation (Conservation Law) can be written as:

Q

n+1i = Q

ni �

�t

�x

(Fni+1/2 � F

ni�1/2)

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Lax-Wendroff (Flux Form)

A flux form of the Lax-Wendroff Method can be defined:

The Lax-Wendroff solution for the advection equation can be written as:

Q

n+1i = Q

ni �

�t

�x

(Fni+1/2 � F

ni�1/2)

F

ni�1/2 =

12c(Qn

i�1 + Q

ni )� 1

2�t

�x

c

2(Qni �Q

ni�1)

Q

ni = Q

ni �

c�t

2�x

(Qni+1 �Q

ni�1) +

12

(c�t)2

(�x)2(Qn

i�1 � 2Q

ni + Q

ni+1)

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Richtmeyer Method

Recall the Lax-Wendroff method as it is often implemented in a 2-step (“Richtmeyer”) form:

Now, we can see that the method explicitly calculates numerical fluxes via the initial half-step.

u

n+1/2j+1/2 =

12(un

j + u

nj+1)�

�t

2�x

(fnj+1 � f

nj )

F

n+1/2j+1/2 = f(un+1/2

j+1/2 )

u

n+1j = u

nj �

�t

�x

(Fn+1/2j+1/2 � F

n+1/2j�1/2 )

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Contents

• Dimensional Splitting• Flux Limiters

EP711 Supplementary MaterialTuesday, February 28, 2012

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Flux Limited MethodsBy defining flux form methods, we can construct special “flux corrected” and “flux limited” methods, which combine the best features of low-order and high-order methods [e.g., Durran, 2010; LeVeque, 2002].

We will focus first on Flux-Limited flux form finite difference methods, using fluxes defined by combinations of low (FL ) and high (FH) order flux approximations.

Fi�1/2 = FLi�1/2 + �i�1/2(FH

i�1/2 � FLi�1/2)

Here, the “flux limiter” is given by phi.

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Calculations for Flux Limiter

The flux limiter function will be dependent on the ratio of solution slopes across an interface between two cells:

The flux limiter value phi is a function of theta...

✓ni+1/2 =

Qni �Qn

i�1

Qni+1 �Qn

i

�(✓ni+1/2)

Let’s investigate this experimentally, then discuss the stability consequences and TVD criteria on Thursday!

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