AMS 691Special 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

– Flows with complex geometry• Mulitphase flows; interface between phases

– Very complex if flow is turbulent– Professors Xaioliln Li, Xiangmin Jiao

– Flows with complex physics• Magnetohydrodunamics (MHD)

– Professor Roman Samulyak• ;Chemistry, combustion, chemical reactions

– James Glimm• Turbulent transport

– James Glimm• ,Phase transitions, material strength and fracture

– Professor Roman Samulyak• Coupling multiple physical models

– Climate studies Xiangmin Jiao, James Glimm, Roman Samulyak• Porous media

– Brent Lindquist– Quantum level modeling; atoms and electrons

• Density functional theory– James Glimm

– Molecular dynamics, biological modeling• Yuefan Deng

– Uncertainty quantification and QMU• Analysis of errors; assurance of accuracy

– 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

– Computer Science Issues• Xianjmin Jiao, Yuefan Deng, James Glimm

– Computational issues in Finance• James Glimm, Xaiolin Li, Andrew Mullhaupt

– Many applications

• 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

• 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

1 1( )

... ; ( ) ...

( )

( ) 0n n

t

U F U

U F U

U F U

U F U

Total Quantity U is conserved

( , ) ( ( , )) 0D D

t

R R

U x t dx F U x t dx

(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 arethe conserved quantities.

Simple case: Burgers’ Equationn = 1, D = 1

2

( , )

( )0

1( )

20

t

t x

u u x t

f uu

x

f u u

u uu

Simpler case: f(u) = aulinear equation (a = const)

0

0

0

0

( , ) ( )

( ) ( , 0)

is given data

t xu a u

u x t u x at

u x u x t

u

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

Space time plot ofcharacteristic curves

Moving discontinuity,plotted u vs. x, moving intime

Simple Equation: Burgers’ Equation

• Ut +(1/2) (U2)x = 0• Ut + 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 ofcharacteristic 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

2

2 2

[ ] jump in quantity across discontinuity

1[ ] = [ ] 0 at moving discontinuity

21

[ ( )] ( ) [ ]2 2 2

2

a a a

s u u

u u u uu u u u u

u us

Weak Solution

2

2

2

2 2

1 all smooth

2

1=

2Choose = ( ) "pillbox"

1= '

2

1 1' ; ' [ ]; ' [ ]

2 2[ ]

t x

t x

u u dxdt

u u dxdt

x st

su u dxdt

u u u u

s u

Compression wave breaking into a shock wave for Burgers equation

Space time plot ofcharacteristic curves.curves meet at the line of discontinuity (a shockwave)

Compressible Fluid Dynamics Euler Equation (1D)

1 1

3 3

2

( )

... ; ( ) ...

( )

( ) 0

mass density; momentum density, = pressure;

1 + = total energy density; = internal energy

2

t

U F U

U m F U

U E F U

U F U

m P

E mv e e

v

F vv P

Ev vP

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

(2 ) (2 ) matrix

acoustic matrix

Governs small amplitude (linear) disturbances

Eigenvalues and eigenvectors of

known by exact formulae (for simple

equations of state), and these are used in s

FA D D

UA

A

ome

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