Finite Difference Solutions For Advection Problems / Hyperbolic PDEs

EP711 Supplementary MaterialThursday, February 2, 2012

Jonathan B. Snively Embry-Riddle Aeronautical University

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Contents

• Review of Conservation Laws• Basis for Finite Difference Solutions• The Lax and Lax-Wendroff Schemes

EP711 Supplementary MaterialThursday, February 2, 2012

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Conservation Laws and Continuity EquationsThis more general advection equation can be considered a “continuity equation” or “conservation law” for a quantity Q that is conserved under all conditions – for example, mass density, momentum, or energy.

@Q

@t+r · (Q~v) = 0

Here, the quantity in parentheses is known as the “flux” f, such that the expression can be written as:

@Q

@t= �r · ~f

Although Q is here a scalar, it does not need to be!

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Conservation of Mass DensityFor a scalar quantity (such as density), the “flux” f depends on Q(x,t), such that the expression can be written as:

@Q

@t= �r · ~f

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:

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Conservation of MomentumAn example of a vector conservation law has already been presented. Momentum equations for each spatial direction (x, y, z for Cartesian coordinates) are here contained in one expression.

@

@t(⇢~v) = �r · (⇢vivj + p�ij)

This expression effectively describes Newton’s law. In one dimension, momentum conservation can be expressed:

@

@t

(⇢v) = � @

@x

(⇢v2 + p)

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Conservation of EnergyEnergy is a scalar quantity (like density):

Here, it is important to define how E relates to other quantities via an equation of state – We will assume an ideal single-constituent gas.

@E

@t= �r · {(E + p)~v}

E = ⇢✏+1

2⇢(~v · ~v)

The internal energy is given by: ✏ = cvT =p

⇢(� � 1)

KineticInternal +

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Model Equations of Motion:We model the atmosphere as a non-rotating, fully-nonlinear, compressible

gas:

⇧⌅

⇧t+⇧ · (⌅�v) = 0 (1)

⇧

⇧t(⌅�v) +⇧ · (⌅�v�v) = �⇧p� ⌅�g (2)

⇧E

⇧t+⇧ ·{ (E + p)�v} = �⌅gvz (3)

where the energy equation and the equation of state for an ideal gas are definedas:

E =p

(� � 1)+

12⌅(�v · �v) (4)

where ⌅ is density, p is pressure, �v is the fluid velocity, along with energy densityE. The Euler equations are coupled with equations for viscosity and thermalconduction:

⇧�v

⇧t= ⇤⇧2�v (5)

⇧T

⇧t= ⇥⇧2T (6)

Mass Density:

Momentum:

Energy:

State:

Viscosity:

Conduction:

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Wave EquationsThe hyperbolic PDEs that we will investigate in this course yield wave equation solutions when manipulated in the appropriate context:

The wave equation can be described instead in terms of two advection equations with opposite speeds:

Physical systems that can be described in this way include Maxwell’s equations...

@

2u

@t

2� c

2 @

2u

@x

2= 0

@2u

@t2� c2r2u = 0

@u

@t

� c

@u

@x

= 0@u

@t

+ c

@u

@x

= 0

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Maxwell’s Equations:

Another familiar example is the solution for waves on a string, or a simple acoustic wave solution. In these cases, two independent variables are incorporated (here, Ey and Bz).

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Dispersion RelationsWave dispersion relations relate the effective wavenumber and frequency of waves within a system under the assumption of Fourier time-harmonic solutions of the form:

The resulting expression can be rewritten as:Which is an ideal dispersion relation for waves having phase velocity and group velocity equal to speed c.

NOTE: c is not the fluid velocity, but a characteristic speed of the physical system

@

2u

@t

2� c

2 @

2u

@x

2= 0

u(x, t) = U

o

e

j(!t�kx)

�!2 + c2k2 = 0

! = ck

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Dispersion RelationsRecall that the dispersion relation for gravity waves indeed quite interesting! And, as we derived it, it also incorporated a solution for acoustic waves (acoustic branch):

Gravity waves are transverse wave motions below the Brunt-Vaisala fre-quency N and the acoustic cut-o� frequency �o.

Neglecting the acoustic terms, a convenient form for the incompressible dis-persion relation relates kx and kz via the wave vector angle � to the horizontal(see Figure) [e.g., Nappo, 2002, p. 32]:

� =kxN

(k2x + k2

z)1/2= N cos �.

k2z =

�2 � �2o

c2s

� �2 �N2

v2�x

As stated, the compressible dispersion relation is:

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Contents

• Review of Conservation Laws• Basis for Finite Difference Solutions• The Lax and Lax-Wendroff Schemes

EP711 Supplementary MaterialThursday, February 2, 2012

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Discretization of Systems in Space and Time

First, let’s discuss notation:

j j+1 j+2j-1j-2

n

n-1

n+1

x-axis domain

t-steppingevolution

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j j+1 j+2j-1j-2

n

n-1

n+1

x-axis domain

t-steppingevolution

Where t=n*∆t, x=i*∆x, y=j*∆y, z=k*∆z

Indices are given by: n, i, j, k (i and j often interchanged...)

Spatial and Temporal Step Sizes are given by ∆t,∆x,∆y,∆z.

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Derivatives on a Mesh

Recall that derivatives can be approximated on a mesh, i.e., the first order centered difference approximation:

dF

dx

⇠ Fj+1 � Fj�1

2�x

j j+1j-1

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Boundary Conditions

For our first numerical experiments we will use the simplest possible boundary conditions: none. Our domains will be periodic, such that any perturbation can propagate out of one domain boundary and back into the other.

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Contents

• Review of Conservation Laws• Basis for Finite Difference Solutions• The Lax and Lax-Wendroff Schemes

EP711 Supplementary MaterialThursday, February 2, 2012

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Lax MethodThe well-known Lax method for solving hyperbolic PDEs is first order accurate, and based on a centered in space difference solution:

j j+1j-1

n

n+1

@u

@t

= �@F

@x

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Lax-Wendroff MethodThe Lax-Wendroff method for solving hyperbolic PDEs is second order accurate, and is often implemented in a 2-step (“Richtmeyer”) form.

j j+1⁄2 j+1j-1⁄2j-1

n+1⁄2

n

n+1Two-stept-steppingevolution

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Lax-Wendroff MethodThe Lax-Wendroff method for solving hyperbolic PDEs is second order accurate, and is often implemented in a 2-step (“Richtmeyer”) form.

This apparently simple approach has numerous pitfalls, many of which can be well-controlled given appropriate choices of time-steps and filtering / stabilizing viscosity.

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Courant-Friedrichs-Lewy Stability Conditions

For finite difference solutions of advection problems, the non-dimensional Courant-Friedrichs-Lewy (CFL) number must be kept below 1 to ensure stability for most schemes.

However, note that for many schemes the behavior may be far more complex than “stable” or “unstable”, exhibiting either dispersion or diffusion.

CFL = c

�t

�x

Solutions stable for:

Here, c is the fastest propagation velocity on the grid (where both wave speeds and advective flows contribute!)

�t �x

c

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