Syllabus

Please note that these schedules are tentative and they may be modified if needed.

Week 1: Basics concepts of fluid flows; conservation laws; history, motivations, challenges, goals, and applications of CFD; Basic concepts for numerical PDEs; differential and integral forms of conservation laws; linear advection & diffusion equations

Week 2: Scalar conservation laws; domain of dependence & influence; shocks & discontinuous solutions; weak solutions; Riemann problems & Rankine-Hugoniot relation; Entropy condition; Numerical methods for scalar conservation laws; solution convergence & truncation errors; numerical stability

Week 3: Numerical solutions for discontinuity; modified equations; numerical dissipations and dispersion errors; Linear hyperbolic systems; characteristics and waves; nonlinear hyperbolic systems; Euler equations, equations of state; entropy

Week 4: Linearization of nonlinear equations; Riemann problems & shocks; Hugoniot locus; genuinely nonlinearity vs. linearly degeneracy; entropy condition; Riemann problems for Euler equations; Riemann fans; Examples of 1D shock tube problems using the FLASH code

Week 5: Conservative vs. non-conservative numerical methods for nonlinear scalar conservation laws; Godunov methods; entropy fix

Week 6: Gas dynamics (nonlinear system of conservation laws) & Euler equations; ideal gas; entropy; Riemann problems for Euler equations; Riemann invariants; Godunov methods for Euler equations; approximate Riemann solvers of Roe, HLL, and HLLC, Rusanov

Week 7: Nonlinear stability; TVD & monotone preserving methods; Godunov Theorem; von Neumann stability analysis; Piecewise-polynomial reconstruction methods for nonlinear scalar conservation laws; slope limiters; low-order methods using first-order Godunov & second-order piecewise linear method (PLM);

Week 8 ~ 9: High-order methods using third-order piecewise-parabolic method (PPM); Essentially no-oscillatory method (ENO); Weighted ENO (WENO);

Week 9 ~ 10: Extension to multi-dimensional problems (2D) using dimensional splitting method and dimensional unsplit method; donor-cell vs. corner-transport-upwind (CTU) methods; von Neumann stability analysis; Boundary conditions; ghost cells; AMR (prolongation & restriction); parallel computing

Week 10 (time permitting): Beyond the basic physics and numerics: 3D, Navier-Stokes equations; MHD, high-energy-density; implicit solvers; Numerical models and algorithms for future high-performance computing architecture; Parallel performance & optimization; scaling; software engineering, code verification & validation