AMS 691
Special Topics in Applied
Mathematics
James Glimm
Department of Applied Mathematics
and Statistics,
Stony Brook University
Brookhaven National Laboratory
0-3 credits
• For 2-3 credits, a term paper is required.
– Pick any ongoing area of CAM research, determine
what the research directions are, and describe current
activities.
– Or pick any result unproven in this course, referred to
some reference
• The course will survey ongoing CAM research
– Guest lectures from other CAM faculty
• Introduction/survey of all CAM research areas
– Some emphasis on turbulent combustion
– Requires significant background material, which will
be surveyed and developed as we progress
• Some details will be omitted, some will be summarized
CAM Research
•
Central themes
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Flows with complex geometry
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Mulitphase flows; interface between phases
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Flows with complex physics
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Magnetohydrodunamics (MHD)
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;Chemistry, combustion, chemical reactions
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Turbulent transport
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,Phase transitions, material strength and fracture
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Coupling multiple physical models
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Porous media
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Very complex if flow is turbulent
Professors Xaioliln Li, Xiangmin Jiao
Professor Roman Samulyak
James Glimm
James Glimm
Professor Roman Samulyak
Climate studies Xiangmin Jiao, James Glimm, Roman Samulyak
Brent Lindquist
Quantum level modeling; atoms and electrons
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Density functional theory
–
James Glimm
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Molecular dynamics, biological modeling
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Uncertainty quantification and QMU
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Yuefan Deng
Analysis of errors; assurance of accuracy
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Verification: is a numerical solution a valid approximation to the mathematical equations
Validation: are the mathematical equations a valid approximation to the physical problem
Uncertainty Quantification: estimate of errors from any and all sources
Quantifice Margins of Uncertainty: numerically designed engineering safety margins for a numerically determined design
James Glimm
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Computer Science Issues
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Computational issues in Finance
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Many applications
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Xianjmin Jiao, Yuefan Deng, James Glimm
James Glimm, Xaiolin Li, Andrew Mullhaupt
• Central Themes
– Mathematical theory, physics modeling and high
performance computing
– Computer science tools to enable effective computing
– Problem specific subject matter
– Required knowledge goes well beyond what is
possible to learn (over the course of your graduate
studies), so as a student, you will learn the parts of
these subjects that you need, for each specific
problem/application.
• Knowledge will be shared among graduate students, to
accelerate the learning process
CAM Research:
Application Areas
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Design of laser fusion; magnetically confined fusion (RS)
Design of new high energy accelerators (RS)
Turbulence, turbulent mixing, turbulent combustion (JG)
Modeling of Scramjet with uncertainty quantification, quantified
margins of uncertainty, verification and validation (JG)
Solar cell design (JG)
Modeling of windmills, parachutes (XL,XJ)
Brittle fracture (RS)
Chemical processing and nuclear power rod fuel separation (JG,XJ)
Flow in porous media; pollution control (XL,BL)
Short term weather forecasting for estimation/optimization of
solar/wind energy (JG)
Porous Media (BL)
Coupling atmosphere and oceans in climate studies (XJ)
Atmospheric modeling (RS,JG)
Compressible/incompressible flows with complex geometry and
physics (XJ)
First Unit: Equations of
Fluid Dynamics
• In some sense, this lecture is an overview of
your main courses for the next two years
• References:
author =
"A. Chorin and J. Marsden",
title =
"A Mathematical Introduction to Fluid Mechnics",
publisher = "Springer Verlag",
address =
"New York--Heidelberg--Berlin",
year =
"2000",
author =
"L. D. Landau and E. M. Lifshitz",
title =
"Fluid Mechanics",
publisher = "Reed Educational and Professional Publishing Ltd",
address =
"London, England",
year =
"1987"
Nonlinear Hyperbolic
Conservation Laws
U1 
 F1 (U ) 
 


U   ...  ; F (U )   ... 
U 
 F (U ) 
 n
 n

 tU  F (U )  0
Total Quantity U is conserved
t
 U ( x, t )dx    F (U ( x, t ))dx  0
RD
RD
(assuming that U vanishes at infinity). Each
component of U is conserved. Fundamental
laws of classical physics are often of this form.
For fluids, mass, momentum and energy are
the conserved quantities.
Simple case: Burgers’ Equation
n = 1, D = 1
u  u ( x, t )
f (u )
t u 
0
x
1 2
f (u )  u
2
 t u  uu x  0
Simpler case: f(u) = au
linear equation (a = const)
 t u  a xu  0
u ( x, t )  u0 ( x  at )
u0 ( x)  u ( x, t  0)
u0 is given data
Linear transport equation
• Ut + aUx = 0
• Solution is constant on lines x = x0 + at.
• These lines are called characteristic
curves.
• Each characteristic line meets initial line, t
= 0 at a unique point .
• Thus solution is defined for all space time:
U(x,t) = U(x-at,0)
• Initial discontinuities in U are preserved in
time, moving with velocity a.
Moving discontinuity for linear transport equation
Moving discontinuity,
plotted u vs. x, moving in
time
Space time plot of
characteristic curves
Simple Equation: Burgers’ Equation
•
•
•
•
•
Ut +(1/2) (U2)x = 0
U t + U Ux = 0
U is a speed, the speed of propagation of information.
Characteristic curves: x = Ut +x0
U = constant on characteristic curve, thus determined by value at t =
0. Characteristic curves are straight lines in 1D space, and time.
Thus solution can be written in closed form by a formula.
– U(x,t) = U0(x-U0t)
– U0(x) = initial data
• Increasing regions of U: characteristic curves spread out, solution
becomes smoother.
• Decreasing regions of U: characteristic curves converge, solution
develops steep gradients, discontinuity, and solution becomes
multivalued.
Moving rarefaction wave for Burgers equation
Space time plot of
characteristic curves
Burgers equation and shock
waves
• [q] = jump in q at discontinuity
• s = speed of moving discontinuity
• Burgers equation interpreted as a
distribution (weak form of equation) at a
discontinuity
– s[u] = [(1/2) u2]
– Solve for s and get formula for solution, with
moving discontinuity (shock wave)
– Extends solution after formation of
discontinuity
[a ]  a  a  jump in quantity across discontinuity
1 2
s[u ] = [ u ]  0 at moving discontinuity
2
u  u
u  u
1 2
2
[ (u  u )] 
(u  u ) 
[u ]
2
2
2
u  u
s
2
Weak Solution
1


2
   ut  2  u  x dxdt all smooth 
1
=  t u  x u 2 dxdt
2
Choose  =  ( x  st ) "pillbox"
1 2

=   '   su  u dxdt
2


1
1 2
2
 '   ;   ' u  [u ];   ' u   [u ]
2
2
 s[u ]
Compression wave breaking into a shock wave for
Burgers equation
Space time plot of
characteristic curves.
curves meet at the line
of discontinuity (a shock
wave)
Compressible Fluid Dynamics
Euler Equation (1D)
U1    
 F1 (U ) 
   


U   ...   m  ; F (U )   ... 
U   E 
 F (U ) 
 3  
 3

 tU  F (U )  0
  mass density; m  momentum density, P = pressure;
1 2
mv +e = total energy density; e = internal energy
2
 v 


F    vv  P 
 Ev  vP 


E
Equation of State (EOS)
• System does not close. P = pressure is an extra unknown; e =
internal energy is defined in terms of E = total energy.
• The equation of state takes any 2 thermodymanic variables and
writes all others as a function of these 2.
• Rho, P, e, s = entropy, Gibbs free energy, Helmholtz free energy are
thermodynamic variables. For example we write P = P(rho,e) to
define the equation of state.
• A simple EOS is the gamma-law EOS.
• Reference:
•
author =
"R. Courant and K. Friedrichs",
•
title =
"Supersonic Flow and Shock Waves",
•
publisher = "Springer-Verlag",
•
address =
"New York",
•
year =
"1967
Entropy
• Entropy = s(rho,e) is a thermodynamic
variable. A fundamental principle of
physics is the decrease of entropy with
time.
– Mathematicians and physicists use opposite
signs here. Confusing!
Analysis of Compressible Euler
Equations
F
 A  (2  D)  (2  D) matrix
U
A  acoustic matrix
Governs small amplitude (linear) disturbances
Eigenvalues and eigenvectors of A
known by exact formulae (for simple
equations of state), and these are used in some
modern numerical schemes
Compressible Fluid Dynamics
Euler Equation
• Three kinds of waves (1D)
• Nonlinear acoustic (sound) type waves: Left or
right moving
– Compressive (shocks); Expansive (rarefactions)
– As in Burgers equation
• Linear contact waves (temperature, and, for fluid
concentrations, for multi-species problems)
– As in linear transport equation
Nonlinear Analysis of the Euler
Equations
• Simplest problem is the Riemann problem in 1D
• Assume piecewise constant initial state, constant for x <
0 and x > 0 with a jump discontinuity at x = 0.
• The solution will have exactly three kinds of waves
(some may have zero strength): left and right moving
“nonlinear acoustic” or “pressure” waves and a contact
discontinuity (across which the temperature can be
discontinuous)
• Exercise: prove this statement for small amplitude waves
(linear waves), starting from the eigenvectors and
eigenvalues for the acoustic matrix A
• Reference: Chorin Marsden