Session Key
L = Lecture
R = Recitation
Q = Quiz
| SES # | TOPICS | KEY DATES |
|---|---|---|
| L1 | Introduction to numerical fluid mechanics: Models to simulations, error types. Approximation and round-off errors. Number representations. Errors of numerical operations. | |
| R1 | Introduction to MATLAB® | |
| L2 | Recursion. Truncation errors, Taylor series and error analysis. Error propagation and estimation. Condition numbers. Roots of non-linear equations. Introduction and bracketing methods. | Problem Set 1 posted |
| L3 | Roots of non-linear equations, bracketing methods: bisection / false position. Open methods: Open-point iteration / Newton-Raphson / secant methods, extension to systems of equations. | |
| R2 | Review: Navier-Stokes equations and their approximations. Conservation laws, material derivative, Reynolds transport theorem, constitutive equations. | |
| L4 | Systems of linear equations: Motivations and plans, direct methods, Gauss elimination. | |
| L5 | Systems of linear equations. Gaussian elimination (special cases, multiple right hand sides). LU decomposition and factorization, pivoting. Error analysis for linear systems. Operations counts. |
Problem Set 1 due Problem Set 2 posted |
| R3 | End of Navier-Stokes review (if needed). Compressible and incompressible flows, vorticity, Euler’s equations, potential flows and (boundary) integral equations. | |
| L6 | Systems of linear equations. Special matrices: LU decompositions, tridiagonal systems, general banded matrices, symmetric, positive-definite matrices. Introduction to iterative methods. | |
| L7 | Systems of linear equations. Iterative methods: Jacobi method, Gauss-Seidel iteration, convergence, successive over-relaxation methods, gradient methods, stop criteria, examples. | |
| R4 | Recitation | |
| L8 | End of systems of linear equations: Gradient methods, preconditioning. Krylov methods. Finite-differences (FD): Classification of partial differential equations (PDEs) and examples, error types and discretization properties. | |
| L9 | FD schemes: Finite difference based on Taylor series for higher order accuracy differences and examples. Taylor tables or method of undetermined coefficients. |
Problem Set 2 due Problem Set 3 posted |
| R5 | Recitation | |
| Q1 | Quiz 1 | |
| L10 | FD schemes: Polynomial approximations (Newton, Lagrange, Hermite, and Pade schemes), iterative improvements and extrapolations, boundary conditions, non-uniform grids, grid refinement. | |
| R6 | Recitation | |
| L11 | Finite-differences: Fourier error analysis, introduction to stability: heuristic, energy and Von Neumann methods, hyperbolic PDEs, characteristics. | |
| L12 | Stability, hyperbolic equations. Revisited, Courant–Friedrichs–Lewy (CFL) condition and Von Neumann stability, elliptic equations revisited and FD schemes. |
Problem Set 3 due Problem Set 4 posted |
| R7 | Recitation | |
| L13 | End of elliptic / hyperbolic equations, special advection schemes (donor cell, flux-corrected transport, weighted essentially non-oscillatory (WENO)), parabolic equations revisited and numerical FD schemes. | |
| L14 | Finite volume methods | |
| R8 | Recitation: Finite volume methods (cont.) | |
| L15 | Methods for unsteady problems. Time-marching methods. Ordinary differential equations (ODEs). Initial value problems (IVPs). Euler’s method. Runge-Kutta methods. | |
| L16 | Time-marching (cont.): Higher order ODEs, stiffness and multistep methods. Solutions of the Navier-Stokes equation, incompressible and compressible. |
Problem Set 4 due Problem Set 5 posted |
| R9 | Recitation | |
| L17 | Solutions of the Navier-Stokes equation: Incompressible and compressible. Pressure-correction, fractional-step. | |
| Q2 | Quiz 2 | |
| R10 | Recitation | |
| L18 | Solutions of the Navier-Stokes equation: Incompressible and compressible. vorticity, artificial compressibility and other methods. | |
| R11 | Recitation | |
| L19 | Grid generation and complex geometries. |
Problem Set 5 due Problem Set 6 posted |
| L20 | Finite volume on complex geometries. Finite element methods: Introduction, fluid applications. | |
| R12 | Recitation | |
| L21 | Finite element methods (cont.): Continuous and discontinuous Galerkin Methods. Spectral methods. | |
| L22 | Inviscid flow equations: Boundary element methods. Panel methods. | Problem Set 6 due |
| R13 | Recitation: Special topics, boundary layer equations, ODEs. Boundary value problems. | |
| L23 | Turbulent flows: Models and numerical simulations. | |
| Final project presentations. |
