Numerical approximation of the inviscid compressible equations of fluid dynamics. Analysis of numerical accuracy, stability, and efficiency. Use of explicit, implicit, and flux split methods. Discussion of splitting, approximate factorization, discrete point, and finite volume approaches. Applications to the solution of simple hyperbolic systems of equations and the Euler equations.
Instructor: A. Ferrante
Examines numerical discretization of the incompressible Navier-Stokes equation; projection method, introduction to turbulence; Reynolds Averaged Navier-Stokes equations; algebraic, one-equation, and two-equation turbulence models; large-eddy simulation; direct numerical simulation; and applications to the numerical solution of laminar and turbulent flows in simple geometries.
Instructor: A. Ferrante
AMATH 507 Calculus of Variations (5)
Necessary and sufficient conditions for a weak and strong extremum. Legendre transformation, Hamiltonian systems. Constraints and Lagrange multipliers. Space-time problems with examples from elasticity, electromagnetics, and fluid mechanics. Sturm-Liouville problems. Approximate methods. Prerequisite: AMATH 351 or MATH 307; MATH 324, 327; recommended: AMATH 402 and AMATH 403 or MATH 428 and 429. Offered: W; odd years.
AMATH 568 Advanced Methods for Ordinary Differential Equations (5)
Survey of practical solution techniques for ordinary differential equations. Linear systems of equations including nondiagonable case. Nonlinear systems; stability phase plane analysis. Asymptotic expansions. Regular and singular perturbations. Recommended: AMATH 402 or equivalent. Offered: W.
AMATH 569 Advanced Methods for Partial Differential Equations (5)
Analytical solution techniques for linear partial differential equations. Discussion of how these arise in science and engineering. Transform and Green’ s function methods. Classification of second-order equations, characteristics. Conservation laws, shocks. Prerequisite: AMATH 403, AMATH 568 or MATH 428 or permission of instructor. Offered: Sp.
AMATH 570 Asymptotic and Perturbation Methods (5)
Asymptotics for integrals, perturbation, and multiple-scale analysis. Singular perturbations: matched asymptotic expansions, boundary layers, shock layers, uniformly valid solutions. Prerequisite: AMATH 567, AMATH 568, AMATH 569, or permission of instructor. Offered: A.
AMATH 572 Introduction to Applied Stochastic Analysis (5)
Introduction to the theory of probability and stochastic processes based on differential equations with applications to science and engineering. Poisson processes and continuous-time Markov processes, Brownian motions and diffusion. Prerequisite: AMATH/STAT 506, AMATH 402, or equivalent knowledge of probability and ordinary differential equations. Offered: Sp; even years.
AMATH 573 Coherent Structures, Pattern Formation and Solitons (5)
Methods for nonlinear partial differential equations (PDEs) leading to coherent structures and patterns. Includes symmetries, conservations laws, stability Hamiltonian and variation methods of PDEs; interactions of structures such as waves or solitons; Lax pairs and inverse scattering; and Painleve analysis. Prerequisite: AMATH 569, or permission of instructor. Offered: A; odd years.
AMATH 574 Conservation Laws and Finite Volume Methods (5)
Theory of linear and nonlinear hyperbolic conservation laws modeling wave propagation in gases, fluids, and solids. Shock and rarefaction waves. Finite volume methods for numerical approximation of solutions; Godunov’ s method and high-resolution TVD methods. Stability, convergence, and entropy conditions. Prerequisite: AMATH 586 or permission of instructor. Offered: W.
AMATH 586 Numerical Analysis of Time Dependent Problems (5)
Numerical methods for time-dependent differential equations, including explicit and implicit methods for hyperbolic and parabolic equations. Stability, accuracy, and convergence theory. Spectral and pseudospectral methods. Prerequisite: AMATH 581 or AMATH 584. Offered: jointly with ATM S 581/MATH 586; Sp.
ESS 524 Numerical Heat and Mass Flow Modeling in the Earth Sciences (3)
Numerical solution of steady and transient advective-diffusion equations describing heat and mass transport processes in Earth Sciences, emphasizing finite-volume methods and their relationship to finite-difference and finite-element methods. Topics include discretization methods; coordinate systems; boundary conditions; accuracy; and stability. Prerequisite: MATH 307; MATH 308 or equivalent, or permission of instructor. Offered: Sp; alternate years.