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AMS 599 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.
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AMS 599Special 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 • 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 Glimm • James • ,Phase transitions, material strength and fracture • Professor Roman Samulyak • Coupling multiple physical models • Climate studies Xiangmin Jiao • Porous media • Brent Lindquist • Quantum level modeling; atoms and electrons • Density functional theory • James Glimm • 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 • 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) • Brittle fracture (RS) • Chemical processing and nuclear power rod fuel separation (JG,XJ) • Flow in porous media; pollution control (XL) • Short term weather forecasting for estimation/optimization of solar/wind energy (JG) • Porous Media (BL) • Coupling atmosphere and oceans in climate studies (XJ) • 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"
Total Quantity U is conserved (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.
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 of characteristic curves Moving discontinuity, plotted u vs. x, moving in time
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 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
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)
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!
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