.. _syllabus:

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Syllabus 
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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