E4301: Numerical Methods for PDE’smspieg/e4301/Lectures/Lecture01_2011.pdf · E4301: Numerical Methods for PDE’s ... • An advanced course in numerical analysis (E6302) ... •

E4301: Numerical Methods for PDE’sTu/Thur 11:00-12:15 327 Mudd

www.ldeo.columbia.edu/~mspieg/e4301

Prof. Marc SpiegelmanDept. of Applied Physics and Applied Math

Dept of Earth and Env. Sciences

http://www.ldeo.columbia.edu/~mspieg/e4301


This Course is Not

• An advanced course in numerical analysis (E6302)

• A programming course. You should have some fluency in a programming language of your choice (for projects) and Matlab for homework assignments.

• Going to be comprehensive: e.g. give you the latest/greatest algorithms for application X (but the project will let you explore any application area of your choice, in the language of your choice).

Course Objectives

• Give both understanding and experience in numerical solution of the most commonly occurring PDEs

• Give you enough analytical tools and examples to understand how it can go badly wrong and why.

• Help you start making intelligent choices of models and methods for solving problems of your choosing

Computational Science Choices

• Choose a Problem


• Choose a Problem• Choose a mathematical Model


• Choose a Problem• Choose a mathematical Model• Choose your PDE’s


Continuous description Lu = f



• Choose a Spatial/Temporal Discretization




• Choose a Spatial/Temporal Discretization• Choose a Functional Discretization




• Choose a Spatial/Temporal Discretization• Choose a Functional Discretization Discrete problem

Auh = fh, F(uh) = 0

2




• Choose a Solver:


Auh = fh, F(uh) = 0

2




• Choose a Solver:• Choose Visualization/Analysis Tools:


Auh = fh, F(uh) = 0

2






Auh = fh, F(uh) = 0

2




Choose Your Software!



Auh = fh, F(uh) = 0

2




Choose Your Software!



Auh = fh, F(uh) = 0

2




• Verification: did you solve the mathematical problem as posed?• Validation: Is your model a good description of reality?

n Dimension

1 Dimension

Choose your Problems/PDE’ssingle physics problems

Basic Problems:

• Di↵usion Problems (Parabolic Equations)

@u

@t

= r· Kru

• Wave Equations (Hyperbolic Equations)

@u

@t

+ c

@u

@x

= 0

@

2u

@t

2+ cr2

u = 0

• Steady State BVPs (Elliptic Equations)

r2u = ⇢

Plus Boundary and initial conditions

3

n Dimension

1 Dimension

Choose your Problems/PDE’ssingle physics problems

Plus Initial and Boundary Conditions!

Basic Problems:

• Di↵usion Problems (Parabolic Equations)

@u

@t

= r· Kru

• Wave Equations (Hyperbolic Equations)

@u

@t

+ c

@u

@x

= 0

@

2u

@t

2+ cr2

u = 0

• Steady State BVPs (Elliptic Equations)

r2u = ⇢

Plus Boundary and initial conditions

3

• Fluid Dynamics (Incompressible Navier Stokes

Equations)

@v

@t

+ v · rv = �rP + ⌫r2v +1

⇢

f

r· v = 0

• Elastic Wave propagation

⇢

@

2u

@t

2= µr2u + (� + µ)r(r· u)

4

Choose your Problems/PDE’smore complex vector valued problems

Choose your Problems/PDE’s“Multi-Physics Problems”

• Thermal Convection (Advection/Di↵usion

of heat + Equations of motion)

@T

@t

+ v · rT = r2T

@v

@t

+ v · rv = �rP + ⌫r2v + RaT g

r· v = 0

• Magma Dynamics (my personal favorite)

D�

Dt

= (1� �)P⇠

+ �/⇢

s

�r·K

µ

rP+P⇠

= r·K

µ

⇥rP

⇤ + �⇢g⇤+�

�⇢

⇢

f

⇢

s

r·V =P⇠

rP

⇤ = r· ⌘⇣rV + rVT

⌘� ��⇢g

5




@T

@t

+ v · rT = r2T

@v

@t

+ v · rv = �rP + ⌫r2v + RaT g

r· v = 0


D�

Dt

= (1� �)P⇠

+ �/⇢

s

�r·K

µ

rP+P⇠

= r·K

µ

⇥rP

⇤ + �⇢g⇤+�

�⇢

⇢

f

⇢

s

r·V =P⇠

rP

⇤ = r· ⌘⇣rV + rVT

⌘� ��⇢g

5

A hybrid FEniCS/PETSc code forInfinite Prandtl Number Thermal Convection



@T

@t

+ v · rT = r2T

@v

@t

+ v · rv = �rP + ⌫r2v + RaT g

r· v = 0


D�

Dt

= (1� �)P⇠

+ �/⇢

s

�r·K

µ

rP+P⇠

= r·K

µ

⇥rP

⇤ + �⇢g⇤+�

�⇢

⇢

f

⇢

s

r·V =P⇠

rP

⇤ = r· ⌘⇣rV + rVT

⌘� ��⇢g

5




@T

@t

+ v · rT = r2T

@v

@t

+ v · rv = �rP + ⌫r2v + RaT g

r· v = 0


D�

Dt

= (1� �)P⇠

+ �/⇢

s

�r·K

µ

rP+P⇠

= r·K

µ

⇥rP

⇤ + �⇢g⇤+�

�⇢

⇢

f

⇢

s

r·V =P⇠

rP

⇤ = r· ⌘⇣rV + rVT

⌘� ��⇢g

5


Plus IC’s, BC’s and constitutive relationships

Non-linear porosity waves(Scott & Stevenson, 1984 Nature, Spiegelman, JFM, 1993, Simpson et al 2009,2010)

Collision of 2, 2D-porosity waves. Instability of 1D waves to 3D waves


Computational Seismology

Weak Form

Choose w,u � V s.t.�

��w · uttdV =

�

��w : ⇥ + M : �w(xs)S(t)

s = 0, st = 0 at t = 0

Note: Free stress Boundary conditions ·n = 0 areautomatically included as natural boundary conditions.

Issue is simply choice of Discrete function space V for testfunctions w and u.

Benchmarked Hybrid FEniCS/PETSc codes (Simpson & Spiegelman, 2010 J. Sci. Comp) using P2-P2 mixed elements with Semi-Lagrangian 2nd-order time stepping.


Choose your Mesh(Spatial/Temporal Discretization)

Uniform cartesian mesh

Staggered mesh(finite Volume methods)


Unstructured Meshes (from FEniCS/Dolfin)

2-D Triangular3-D Tetrahedral


Unstructured Hexahedral Meshes (from Cubit (Sandia)FEniCS/Dolfin)


Semi-Structured

Forest of Oct-tree Mesh (K Burstedde, ICES, UT Austin)


Unstructured Meshes (a toilet)

Choose a functional discretization

• Truncated Taylor series (Finite Difference)

• Linear combination of N basis functions

• local compact basis (FEM, FVEM methods)

• Global basis functions (Spectral methods)

• High order local bases (Spectral Elements)

u

h(x) =NX

i=1

w

i

�i

(x)

6

Transformation from Continuous to Discrete problem

• Introduces Truncation Error!

• Need to analyze both stability and convergence.

• Transforms problem to discrete problems

• Requires efficient solvers for algebraic systems of linear and non-linear equations

Auh = fh, F(uh) = 0

2

Choose your solvers• Whole families of methods for large sparse

systems

• Sparse direct (x=A\b)

• Simple iterative (Jacobi, SOR)

• Advanced Iterative methods:

• Krylov methods (CG, GMRES)

• Multi-grid methods

• Combined Preconditioned methods

• Newton-Krylov for nonlinear problems

• Physics based block-preconditioners (all of the above)

Good software helps making choices (and changing your mind) easier

Choose your Software!



• Basic Programming Languages and tools

• C,C++,Python,Fortran

• Editors, Debuggers, Version Control (hg,svn etc)






• Higher level programming languages and computational environments

• Matlab, Mathematica, SciPy,Octave,Sage,R








• Advanced computational Libraries!

• Parallel Linear Algebra: PETSc, Trilinos

• Finite Elements: FEniCS, deal.ii, etc








• Advanced computational Libraries!

• Parallel Linear Algebra: PETSc, Trilinos

• Finite Elements: FEniCS, deal.ii, etc

• Visualization/Analysis software

• Matlab, SciPy, Paraview, Sage...

Course Objective

• Give Guidance and experience to start making intelligent choices (and enough tools and breadth to change your mind)

Course Logistics• All information on web-site, http://

www.ldeo.columbia.edu/~mspieg/e4301

• Link also in Courseworks

• Principal approach is lectures, problem sets (~6) and projects.

• Projects can be collaborative up to 3/group

• Fill out the User Survey so I can gauge background and interests





http://www.ldeo.columbia.edu/~mspieg/e4301/background_form.html

http://www.ldeo.columbia.edu/~mspieg/e4301/background_form.html

VolcanosEarthquakes

A (complicated) exampleUnderstanding the dynamics of Subduction Zones

VolcanosEarthquakes

A (complicated) exampleUnderstanding the dynamics of Subduction Zones

Choices• Problem: Subduction Zones

• why? Earthquakes, Volcanos (cool!)

• PDEs: Magma Dynamics (McKenzie 84, Spiegelman 93, Katz 2007, Simpson 2010)

• Discretization:

• mesh: Unstructured Triangles

• elements: P1-P1 (velocity, Temperature), P2-P2 Porosity Pressure

• Time stepping: Semi-Lagrangian Crank Nicolson

• Solvers: Newton-Krylov with Physics-based block-preconditioned GMRES

• Software: SEPRAN, hybrid FEniCS/PETSc w/ Paraview

An Example Solution




Computational Seismology

Weak Form

Choose w,u � V s.t.�

��w · uttdV =

�

��w : ⇥ + M : �w(xs)S(t)

s = 0, st = 0 at t = 0

Note: Free stress Boundary conditions ·n = 0 areautomatically included as natural boundary conditions.

Issue is simply choice of Discrete function space V for testfunctions w and u.

Benchmarked Hybrid FEniCS/PETSc codes (Simpson & Spiegelman, 2010 J. Sci. Comp) using P2-P2 mixed elements with Semi-Lagrangian 2nd-order time stepping.


Comparison to Data0

200

Depth

, km

5

5

5

10

15

0 1 2 3 4 5 6 7 8 9 10 11 12 130.00 0.04 0.08

0 100 200 300 400

Distance, kmSW NE

0

200

100 100

0 100 200 300 400

Distance, kmSW NE

Vp/Vs1000/Qs

low permeability high permeability

.

Comparison to Data0

200

Depth

, km

5

5

5

10

15

0 1 2 3 4 5 6 7 8 9 10 11 12 130.00 0.04 0.08

0 100 200 300 400

Distance, kmSW NE

0

200

100 100

0 100 200 300 400

Distance, kmSW NE

Vp/Vs1000/Qs

low permeability high permeability

.

132.9 ± 4.1 km(Syracuse & Abers)

The Future? GuMBi

• A prototype Model Building interface

• encapsulates the full set of choices in a single options file

• Built on existing Libraries: FEniCS, PETSc, spud

• Allows complete model building for custom compiled applications through code generation.

• FEM for General, coupled PDE’s on unstructured meshes (from FEniCS)

• Physics based, flexible block-preconditioners ( for non-linear multi-physics problems (built on PETSc SNES)

• Self documenting schema for options files (from spud)

The Future? GuMBi

Course Objectives

• Basic tools and insights to understand and use these and other computational tools

• Some real practical experience in solving a problem of your own.

By the end of this course, you should have:

Documents

E4301: Numerical Methods for PDE’smspieg/e4301/Lectures/Lecture01_2011.pdf · E4301: Numerical Methods for PDE’s ... • An advanced course in numerical analysis (E6302) ... •