Numerical Fluid Dynamics – Lecture Notes

Numerical Fluid Dynamics lecture notes

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Chapter 1 Equations of hydrodynamics
1.1 Basic quantities
1.2 Ideal gases
1.2.1 Most used variables in numerical hydrodynamics
1.3 The first law of thermodynamics
1.4 The Euler equations: the equations of motion of the gas
1.4.1 Conservation of mass
1.4.2 Conservation of momentum
1.4.3 Conservation of energy
1.5 Lagrange form of the hydrodynamics equations
1.6 Properties of the hydrodynamic equations
1.7 Sound waves
1.8 Viscosity
1.9 Shock waves

Chapter 2 Hyperbolic equations
2.1 The simplest form of a hyperbolic equation: advection
2.2 Advection with space-dependent velocity
2.3 Advection of a conserved quantity with space-dependent velocity
2.4 Flux-form of conservation equation and the Jacobian
2.5 Coupled set of equations
2.6 The wave equation in vector notation: eigenvalues and eigenvectors
2.7 Hyperbolic sets of equations: the linear case with constant Jacobian
2.8 Boundary conditions (I)
2.9 Hyperbolic sets of equations: the linear case with variable Jacobian
2.10 Boundary conditions (II)
2.11 Hyperbolic equations versus elliptic equations
2.12 Hyperbolic equations: the non-linear case
2.13 Example of non-linear conservation equation: Burger’s equation
2.14 Isothermal hydrodynamic equations
2.15 Non-isothermal hydrodynamic equations
2.16 Traffic flow equations
2.17 Hyperbolic equations in 2-D and 3-D

Chapter 3 Advection algorithms I. The basics
3.1 Prelude: Numerical integration of an ordinary differential equation
3.2 Numerical spatial derivatives
3.3 Some first advection algorithms
3.4 Numerical diffusion
3.5 Courant-Friedichs-Lewy condition
3.6 Local truncation error and order of the algorithm
3.7 Lax-Richtmyer stability analysis of numerical advection schemes
3.8 Von Neumann stability analysis of numerical advection schemes
3.9 Phase errors and Godunov’s Theorem
3.10 Computer implementation (adv-1): grids and arrays

Chapter 4 Advection algorithms II. Flux conservation, subgrid models and flux limiters
4.1 Flux conserving formulation: the principles
4.2 Donor-cell advection
4.3 Piecewise linear schemes
4.4 Slope limiters: non-linear tools to prevent overshoots
4.5 Flux limiters
4.6 Overview of algorithms
4.7 Computer implementation (adv-2): cell interfaces and ghost cells

Chapter 5
Classic hydrodynamics solvers
5.1 Basic approach
5.2 A simple 1-D hydrodynamics algorithm
5.3 Boundary conditions, ghost cells
5.4 Hydrodynamics with ghost cells
5.5 Now including the energy equation
5.6 Shock waves and the Von Neumann – Richtmyer artificial viscosity
5.7 Odd-even decoupling
5.8 Staggered grids
5.9 External gravity force
5.10 Alternative methods for the energy equation
5.11 2-D/3-D Hydrodynamics
5.12 Practical matters: input and output of data

Chapter 6 Riemann solvers I

6.1 Simple waves, integral curves and Riemann invariants
6.2 Riemann problems
6.3 Riemann problems for the equations of hydrodynamics
6.4 Godunov’s method
6.5 Godunov for linear hyperbolic problems: a characteristic solver

Chapter 7 Riemann solvers II

7.1 Roe’s linearized Riemann solver
7.2 Properties of the Roe solver
7.3 The HLL family of solvers
7.4 Source extrapolation methods
7.5 Employing slope limiters before the Riemann solver step
7.6 The PPM Method
7.7 Code testing: the Sod shock tube tests

Chapter 8 Coordinate systems and gridding techniques

8.1 Hydrodynamics in the polar coordinate system
8.2 Adaptive Mesh Refinement
8.3 Unstructured grids, or partly structured grids

Chapter 9 Implicit integration, incompressible flows

9.1 Simple example: an ordinary differential equation
9.2 A 1-D diffusion equation
9.3 Diffusion equation in 2-D and 3-D: a prelude
9.4 Iterative solvers for sparse matrix equations
9.5 Incompressible fluid equations

Chapter 10 Smooth Particle Hydrodynamics Solvers

10.1 Lagrange equations of hydrodynamics for SPH
10.2 The SPH Kernel
10.3 Some expressions in SPH-form
10.4 The SPH equations of motion
10.6 Shocks: artificial viscosity
10.7 Some thoughts about SPH
10.8 The code GADGET-2

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2 thoughts on “Numerical Fluid Dynamics – Lecture Notes

  1. Karthi

    This material is very helpful,
    written in a great manner, and has solutions to the numerical programming.


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