Clawpack Tutorial Part 2 - University of Washington

Clawpack TutorialPart 2

Randall J. LeVequeApplied Mathematics

University of Washington

Slides posted atwww.clawpack.org/links/tutorials

Randy LeVeque Clawpack Tutorial at the IMA, October 2010

http://www.clawpack.org/links/tutorials

Outline

• Makefiles• Specifying run-time parameters• Specifying boundary conditions• Riemann solvers


Application Makefile

Documentation: www.clawpack.org/doc/makefiles.htmlFor example, see $CLAW/apps/acoustics/1d/example2/Makefile

Several variables are set, e.g.where to find setrun function and where to put output:

CLAW_setrun_file = setrun.pyCLAW_OUTDIR = _output

where to find setplot function and where to put plots:

CLAW_setplot_file = setplot.pyCLAW_PLOTDIR = _plots

Usually these do not need to be changed.


http://kingkong.amath.washington.edu/clawpack/doc/makefiles.html

http://kingkong.amath.washington.edu/clawpack/trunk/apps/acoustics/1d/example2/Makefile.html

Application Makefile (cont.)

List of local Fortran files:

CLAW_SOURCES = \driver.f \qinit.f \rp1.f \setprob.f

List of library files:

# Clawpack library to be used:CLAW_LIB = $(CLAW)/clawpack/1d/lib

CLAW_LIBSOURCES = \$(CLAW_LIB)/claw1ez.f \$(CLAW_LIB)/bc1.f \etc.


Application Makefile (cont.)

Ends with...

# Include Makefile containing standard# definitions and make options:CLAWMAKE = $(CLAW)/util/Makefile.commoninclude $(CLAWMAKE)

The file $CLAW/util/Makefile.common contains rules for varioustargets.

For possible targets, type

$ make help

Documentation: www.clawpack.org/doc/makefiles.html


http://kingkong.amath.washington.edu/clawpack/trunk/util/Makefile.common

http://kingkong.amath.washington.edu/clawpack/doc/makefiles.html

Setting runtime parameters

The file setrun.py contains a function setrunthat returns an object rundata of class ClawRunData.

Never need to write from scratch...Modify an existing example!

Don’t need to know much if anything about Python!

Lots of comments in the sample versions.

Documentation: www.clawpack.org/doc/setrun.html

For example, see $CLAW/apps/acoustics/1d/example2/setrun.py


http://kingkong.amath.washington.edu/clawpack/doc/setrun.html

http://kingkong.amath.washington.edu/clawpack/trunk/apps/acoustics/1d/example2/setrun.py.html

Copying and modifying an example

Find an example similar to the one you want to create and copythe directory, e.g.

$ cd $CLAW/apps/acoustics/1d$ cp -r example2 $CLAW/myclaw/newexample

Warning: If you have a Subversion copy of Clawpack...This will copy .svn subdirectory too.Better way:

$ cd $CLAW/apps/acoustics/1d$ svn export example2 $CLAW/myclaw/newexample

Documentation: www.clawpack.org/doc/newapp.html


http://kingkong.amath.washington.edu/clawpack/doc/newapp.html

Copying and modifying an example

Find an example similar to the one you want to create and copythe directory, e.g.

$ cd $CLAW/apps/acoustics/1d$ cp -r example2 $CLAW/myclaw/newexample

Warning: If you have a Subversion copy of Clawpack...This will copy .svn subdirectory too.Better way:

$ cd $CLAW/apps/acoustics/1d$ svn export example2 $CLAW/myclaw/newexample

Documentation: www.clawpack.org/doc/newapp.html


http://kingkong.amath.washington.edu/clawpack/doc/newapp.html

The Riemann problem

The Riemann problem consists of the hyperbolic equationunder study together with initial data of the form

q(x, 0) ={ql if x < 0qr if x ≥ 0

Piecewise constant with a single jump discontinuity from ql toqr.

The Riemann problem is fundamental to understanding• The mathematical theory of hyperbolic problems,• Godunov-type finite volume methods

Why? Even for nonlinear systems of conservation laws, theRiemann problem can often be solved for general ql and qr, andconsists of a set of waves propagating at constant speeds.


Linear hyperbolic systems

Linear system of m equations: q(x, t) ∈ lRm for each (x, t) and

qt +Aqx = 0, −∞ < x,∞, t ≥ 0.

A is m×m with eigenvalues λp and eigenvectors rp,for p = 1, 2, , . . . , m:

Arp = λprp.

Combining these for p = 1, 2, , . . . , m gives

AR = RΛ

where

R = [r1 r2 · · · rm], Λ = diag(λ1, λ2, . . . , λm).

The system is hyperbolic if the eigenvalues are real andR is invertible. Then A can be diagonalized:

R−1AR = Λ


Linear hyperbolic systems

Let R be matrix of right eigenvectors and v(x, t) = R−1q(x, t).Multiply system qt +Aqx = 0 by R−1 on left to obtain

R−1qt +R−1ARR−1qx = 0

Since R−1AR = Λ, this diagonalizes the system:

wt + Λwx = 0.

This is a system of m decoupled advection equations

wpt + λpwp

x = 0.

Sowp(x, t) = wp(x− λpt, 0)

where w(x, 0) = R−1q(x, 0) = R−1η(x).


Linear acoustics

Example: Linear acoustics in a 1d gas tube

q =[pu

]p(x, t) = pressure perturbationu(x, t) = velocity

Equations:

pt + κux = 0 Change in pressure due to compressionρut + px = 0 Newton’s second law, F = ma

where K = bulk modulus, and ρ = unperturbed density of gas.

Hyperbolic system:[pu

]t

+[

0 κ1/ρ 0

] [pu

]x

= 0.


Linear acoustics

[pu

]t

+[

0 κ1/ρ 0

] [pu

]x

= 0.

This has the form qt +Aqx = 0 with

eigenvalues: λ1 = −c, λ2 = +c,

where c =√κ/ρ = speed of sound.

eigenvectors: r1 =[−Z1

], r2 =

[Z1

]where Z = ρc =

√ρκ = impedance.

R =[−Z Z1 1

], R−1 =

12Z

[−1 Z1 Z

].


Riemann Problem

Special initial data:

q(x, 0) ={ql if x < 0qr if x > 0

Example: Acoustics with bursting diaphram (ul = ur = 0)

Pressure:

Acoustic waves propagate with speeds ±c.Randy LeVeque Clawpack Tutorial at the IMA, October 2010

Riemann Problem


q(x, 0) ={ql if x < 0qr if x > 0


Pressure:


Riemann Problem


q(x, 0) ={ql if x < 0qr if x > 0


Pressure:


Riemann Problem


q(x, 0) ={ql if x < 0qr if x > 0


Pressure:


Riemann Problem for acoustics

ql = w1l r

1 + w2l r

2

qr = w1rr

1 + w2rr

2

Thenqm = w1

rr1 + w2

l r2

So the wavesW1 andW2 are eigenvectors of A:

W1 = qm − ql = (w1r − w1

l )r1

W2 = qr − qm = (w2r − w2

l )r2.


Riemann Problem for acoustics

Waves propagating in x–t space:

Left-going waveW1 = qm − ql andright-going waveW2 = qr − qm are eigenvectors of A.


Godunov’s Method for qt + f(q)x = 0

Then either:

1. Compute new cell averages by integrating over cell at tn+1,

2. Compute fluxes at interfaces and flux-difference:

Qn+1i = Qni −

∆t∆x

[Fni+1/2 − Fni−1/2]

3. Update cell averages by contributions from all waves entering cell:

Qn+1i = Qni −

∆t∆x

[A+∆Qi−1/2 +A−∆Qi+1/2]

where A±∆Qi−1/2 =m∑i=1

(spi−1/2)±Wpi−1/2.



Then either:



Qn+1i = Qni −

∆t∆x

[Fni+1/2 − Fni−1/2]


Qn+1i = Qni −

∆t∆x

[A+∆Qi−1/2 +A−∆Qi+1/2]


(spi−1/2)±Wpi−1/2.



Then either:



Qn+1i = Qni −

∆t∆x

[Fni+1/2 − Fni−1/2]


Qn+1i = Qni −

∆t∆x

[A+∆Qi−1/2 +A−∆Qi+1/2]


(spi−1/2)±Wpi−1/2.


Wave-propagation viewpoint

For linear system qt +Aqx = 0, the Riemann solution consists of

wavesWp propagating at constant speed λp.λ2∆t

W1i−1/2

W1i+1/2

W2i−1/2

W3i−1/2

Qi −Qi−1 =m∑

p=1

αpi−1/2r

p ≡m∑

p=1

Wpi−1/2.

Qn+1i = Qn

i −∆t∆x[λ2W2

i−1/2 + λ3W3i−1/2 + λ1W1

i+1/2

].


CLAWPACK Riemann solver

The hyperbolic problem is specified by the Riemann solver• Input: Values of q in each grid cell

• Output: Solution to Riemann problem at each interface.• WavesWp ∈ lRm, p = 1, 2, . . . , Mw

• Speeds sp ∈ lR, p = 1, 2, . . . , Mw,

• Fluctuations A−∆Q, A+∆Q ∈ lRm

Note: Number of waves Mw often equal to m (length of q),but could be different (e.g. HLL solver has 2 waves).

Fluctuations:

A−∆Q = Contribution to cell average to left,A+∆Q = Contribution to cell average to right

For conservation law, A−∆Q+A+∆Q = f(Qr)− f(Ql)



The hyperbolic problem is specified by the Riemann solver• Input: Values of q in each grid cell

• Output: Solution to Riemann problem at each interface.• WavesWp ∈ lRm, p = 1, 2, . . . , Mw

• Speeds sp ∈ lR, p = 1, 2, . . . , Mw,

• Fluctuations A−∆Q, A+∆Q ∈ lRm

Note: Number of waves Mw often equal to m (length of q),but could be different (e.g. HLL solver has 2 waves).

Fluctuations:

A−∆Q = Contribution to cell average to left,A+∆Q = Contribution to cell average to right

For conservation law, A−∆Q+A+∆Q = f(Qr)− f(Ql)



Inputs to rp1 subroutine:

ql(i,1:m) = Value of q at left edge of ith cell,

qr(i,1:m) = Value of q at right edge of ith cell,

Warning: The Riemann problem at the interface between cellsi− 1 and i has left state qr(i-1,:) and right state ql(i,:).

rp1 is normally called with ql = qr = q,but designed to allow other methods:



Outputs from rp1 subroutine:

for system of m equationswith mw ranging from 1 to Mw =# of waves

s(i,mw) = Speed of wave # mw in ith Riemann solution,

wave(i,1:m,mw) = Jump across wave # mw,

amdq(i,1:m) = Left-going fluctuation, updates Qi−1

apdq(i,1:m) = Right-going fluctuation, updates Qi


Boundary conditions and ghost cells

In each time step, the data in cells 1 to N = mx is used todefine ghost cell values in cells outside the physical domain.

The wave-propagation algorithm is then applied on theexpanded computational domain, solving Riemann problems atall interfaces.

The data is extended depending on the physical boundaryconditons.


Boundary conditions

....Q−1 Q0 Q1 Q2 QN QN+1 QN+2

x1/2

x = a

xN+1/2

x = b

Periodic:

Qn−1 = Qn

N−1, Qn0 = Qn

N , QnN+1 = Qn

1 , QnN+2 = Qn

2

Extrapolation (outflow):

Qn−1 = Qn

1 , Qn0 = Qn

1 , QnN+1 = Qn

N , QnN+2 = Qn

N

Solid wall:For Q0 : p0 = p1, u0 = −u1,

For Q−1 : p−1 = p2, u−1 = −u2.


Boundary conditions

....Q−1 Q0 Q1 Q2 QN QN+1 QN+2

x1/2

x = a

xN+1/2

x = b

Periodic:

Qn−1 = Qn

N−1, Qn0 = Qn

N , QnN+1 = Qn

1 , QnN+2 = Qn

2


Qn−1 = Qn

1 , Qn0 = Qn

1 , QnN+1 = Qn

N , QnN+2 = Qn

N


For Q−1 : p−1 = p2, u−1 = −u2.


Boundary conditions

....Q−1 Q0 Q1 Q2 QN QN+1 QN+2

x1/2

x = a

xN+1/2

x = b

Periodic:

Qn−1 = Qn

N−1, Qn0 = Qn

N , QnN+1 = Qn

1 , QnN+2 = Qn

2


Qn−1 = Qn

1 , Qn0 = Qn

1 , QnN+1 = Qn

N , QnN+2 = Qn

N


For Q−1 : p−1 = p2, u−1 = −u2.


Setting BCs in Clawpack

In setrun.py, at each boundary(xlower, xupper, ylower, yupper)

must specify for example:

clawdata.mthbc_xlower = 3 # solid wallclawdata.mthbc_xupper = 1 # extrapolation

Set to 2 at both boundaries for periodic.

Set to 0 if you want to impose something special.

In this case need to copy $CLAW/clawpack/1d/lib/bc1.f toapplication directory and modify (along with Makefile).


http://kingkong.amath.washington.edu/clawpack/trunk/clawpack/1d/lib/bc1.f.html

Setting BCs in Clawpack

In setrun.py, at each boundary(xlower, xupper, ylower, yupper)

must specify for example:

clawdata.mthbc_xlower = 3 # solid wallclawdata.mthbc_xupper = 1 # extrapolation

Set to 2 at both boundaries for periodic.

Set to 0 if you want to impose something special.

In this case need to copy $CLAW/clawpack/1d/lib/bc1.f toapplication directory and modify (along with Makefile).


http://kingkong.amath.washington.edu/clawpack/trunk/clawpack/1d/lib/bc1.f.html

Extrapolation boundary conditions

If we set Q0 = Q1 then the Riemann problem at x1/2 has zerostrength waves:

Q1 −Q0 =W11/2 +W2

1/2

So in particular the incoming waveW2 has strength 0.

The outgoing wave perhaps should have nonzero magnitude,but it doesn’t matter since it would only update ghost cell.

Ghost cell value is reset at the start of each time step byextrapolation.

In 2D or 3D, extrapolation in normal direction is not perfect butworks quite well.




Q1 −Q0 =W11/2 +W2

1/2








Q1 −Q0 =W11/2 +W2

1/2







Examples:

• One-dimensional acoustics:$CLAW/book/chap3/acousimple$CLAW/apps/acoustics/1d/example2

• Tsunami propagation:$CLAW/apps/tsunami/chile2010

• Seismic waves in a half-space on following slides,


http://kingkong.amath.washington.edu/clawpack/trunk/book/chap3/acousimple/_plots/_PlotIndex.html

http://kingkong.amath.washington.edu/clawpack/trunk/apps/acoustics/1d/example2/_plots/_PlotIndex.html

http://kingkong.amath.washington.edu/clawpack/trunk/apps/tsunami/chile2010/_plots/_PlotIndex.html

Equations of linear elasticity (in 2d)

σ11t − (λ+ 2µ)ux − λvy = 0

σ22t − λux − (λ+ 2µ)vy = 0

σ12t − µ(vx + uy) = 0

ρut − σ11x − σ12

y = 0

ρvt − σ12x − σ22

y = 0

where λ(x, y) and µ(x, y) are Lamé parameters.

This has the form qt +Aqx +Bqy = 0.

The matrix (A cos θ +B sin θ) has eigenvalues −cp, −cs, 0, cs, cp

P-wave (dilatational) speed: cp =√

λ+2µρ

S-wave (shear) speed: cs =√

µρ


Seismic waves in layered earth

Layers 1 and 3: ρ = 2, λ = 1, µ = 1, cp ≈ 1.2, cs ≈ 0.7

Layer 2: ρ = 5, λ = 10, µ = 5, cp = 2.0, cs = 1

Impulse at top surface at t = 0.

Solved on uniform Cartesian grid (600× 300).

Cell average of material parameters used in each finite volumecell.

Extrapolation at computational boundaries.


Seismic wave in layered medium

Red = div(u) [P-waves], Blue = curl(u) [S-waves]





























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Clawpack Tutorial Part 2 - University of Washington