Contents

• List of Figures
• List of Tables
• 1 Introduction of the equations of fluid dynamics
  • 1.1 Presentation of the Euler equations
    • 1.1.1 The Euler equations in differential form (vectors)
    • 1.1.2 The Euler equations in differential form (components)
    • 1.1.3 The solution of the Euler equations
    • 1.1.4 The prototype numerical solution of the Euler equations
  • 1.2 Derivation of the Euler equations
    • 1.2.1 Possible basic quantities
    • 1.2.2 Choice of basic quantities
    • 1.2.3 Fluxes through surface
    • 1.2.4 Changes of the conserved quantities
    • 1.2.5 Euler equations in integral form
    • 1.2.6 Euler equations: from integral to differential form I
    • 1.2.7 Euler equations: from integral to differential form II
  • 1.3 Extensions of the Euler equations
    • 1.3.1 Hydrodynamics equations including viscosity
    • 1.3.2 Hydrodynamics equations including gravity
    • 1.3.3 Hydrodynamics equations including magnetic fields
    • 1.3.4 Hydrodynamics equations including radiation
  • 1.4 Radiation hydrodynamics of stellar atmospheres
    • 1.4.1 Conditions in stellar atmospheres
    • 1.4.2 Stationary case: assumptions
    • 1.4.3 Stationary case: results
    • 1.4.4 Static case
    • 1.4.5 Coupling of radiation transport and hydrodynamics
  • 1.5 Euler equations as hyperbolic system
    • 1.5.1 1D Euler equations in conservation form
    • 1.5.2 Quasi-linear system
    • 1.5.3 Hyperbolic system
    • 1.5.4 Eigenvalues for the Euler equations
• 2 The one-dimensional linear advection equation
  • 2.1 Introduction of the linear advection equation
    • 2.1.1 Linear advection as special case: density and momentum
    • 2.1.2 Linear advection as special case: total energy
    • 2.1.3 Linear advection as special case
    • 2.1.4 Analytic solution of the linear advection equation
    • 2.1.5 Solution along characteristic curves
  • 2.2 Naive numerics: discretization attempts
    • 2.2.1 Simple ODE: discretization
    • 2.2.2 Simple ODE: examples
    • 2.2.3 Simple ODE: remarks
    • 2.2.4 Parabolic PDE: heat equation
    • 2.2.5 Parabolic PDE: discretization
    • 2.2.6 Parabolic PDE: stability
    • 2.2.7 Parabolic PDE: example
    • 2.2.8 Linear advection equation: discretization
    • 2.2.9 Linear advection equation: crash
    • 2.2.10 Linear advection equation: the lesson
  • 2.3 Basic concepts
    • 2.3.1 Discretization in space: wishlist
    • 2.3.2 Discretization in space by finite differences
    • 2.3.3 Discretization in space by finite volumes
    • 2.3.4 Discretization in space by other methods
    • 2.3.5 Discretization in space: grids
    • 2.3.6 Integral form and weak solution
    • 2.3.7 Integral form and flux centering
    • 2.3.8 Centering of quantities, fluxes, and differences
    • 2.3.9 Update formula in conservation form
    • 2.3.10 Stencil diagrams
    • 2.3.11 Stencil diagrams: spatial centering
    • 2.3.12 Stencil diagrams: centering in time
    • 2.3.13 CFL condition
    • 2.3.14 Truncation error
    • 2.3.15 Consistency - stability - convergence
    • 2.3.16 Derivations of donor cell scheme
    • 2.3.17 Further concepts
  • 2.4 Examples
    • 2.4.1 Parameter of the following examples
    • 2.4.2 Naive FTCS scheme
    • 2.4.3 Implicit centered scheme
    • 2.4.4 BTCS scheme
    • 2.4.5 Donor cell (FTBS) scheme
    • 2.4.6 FTFS scheme
    • 2.4.7 Lax-Friedrichs scheme
    • 2.4.8 Lax-Wendroff scheme
    • 2.4.9 Beam-Warming scheme
    • 2.4.10 Fromm scheme
  • 2.5 Analysis of schemes
    • 2.5.1 Overshoot
    • 2.5.2 Artificial viscosity
    • 2.5.3 Modified equation for the FTFS scheme I
    • 2.5.4 Modified equation for the FTFS scheme II
    • 2.5.5 Modified equation for the FTBS scheme
    • 2.5.6 Linear stability analysis of original PDE
    • 2.5.7 Linear stability analysis: use
    • 2.5.8 Linear stability analysis of naive FTCS scheme
    • 2.5.9 Linear stability analysis of donor cell (FTBS) scheme
    • 2.5.10 Linear stability analysis: remarks
  • 2.6 Non-linear schemes
    • 2.6.1 Godunov's idea
    • 2.6.2 Monotonicity
    • 2.6.3 Flux of PLM schemes
    • 2.6.4 Examples: PLM: slopes with linear parameter dependence
    • 2.6.5 PLM: slope-limiter
    • 2.6.6 PLM scheme with Minmod slope-limiter
    • 2.6.7 PLM scheme with vanLeer slope-limiter
    • 2.6.8 PLM scheme with Superbee slope-limiter
    • 2.6.9 PPM scheme
    • 2.6.10 WENO scheme
    • 2.6.11 Scheme with Superbee slope, only boundary value used
    • 2.6.12 Further improvements
• 3 Non-linear advection: Burgers' equation
  • 3.1 Introduction of Burgers' equation
    • 3.1.1 Viscous and inviscid Burgers' equation
    • 3.1.2 Solution along characteristic curves
    • 3.1.3 Compression waves and shocks
    • 3.1.4 Shock speed I
    • 3.1.5 Shock speed II
    • 3.1.6 Expansion waves
    • 3.1.7 Similarity solutions I
    • 3.1.8 Similarity solutions II
    • 3.1.9 Classification of Riemann problems
  • 3.2 Numerical examples
    • 3.2.1 Velocity at cell boundary
    • 3.2.2 Flux-splitting
    • 3.2.3 Example: small-amplitude wave
    • 3.2.4 Example: Gaussian and conservativity
    • 3.2.5 Lax-Wendroff theorem
    • 3.2.6 Example: step-function and conservativity
    • 3.2.7 Example: expansion shock and rarefaction fan
    • 3.2.8 Entropy production by artificial viscosity
    • 3.2.9 Entropy fix
    • 3.2.10 Concepts from the linear world
    • 3.2.11 Transformation of shock speed formula
    • 3.2.12 Second-order extension of flux formula
• 4 One-dimensional non-linear hydrodynamics
  • 4.1 New Difficulties
    • 4.1.1 A coupled non-linear system
    • 4.1.2 Positivity of density
    • 4.1.3 Positivity of pressure
    • 4.1.4 Where is upwind?
  • 4.2 The Riemann problem for the 1D Euler equations
    • 4.2.1 The Riemann problem
    • 4.2.2 Rankine-Hugoniot conditions
    • 4.2.3 Example: Sod shock tube
    • 4.2.4 Jumps in the solution of the Sod shock tube problem
  • 4.3 Riemann solvers
    • 4.3.1 Godunov-type schemes
    • 4.3.2 Riemann solvers and higher-order schemes
  • 4.4 Approximate (linear) Riemann solvers
    • 4.4.1 Linearized flux
    • 4.4.2 Linearized Riemann solver
    • 4.4.3 Examples of schemes
    • 4.4.4 Example: Sod shock tube with Roe solver, constant
    • 4.4.5 Example: Sod shock tube with Roe solver, Minmod slope
    • 4.4.6 Example: Sod shock tube with Roe solver, vanLeer slope
    • 4.4.7 Example: Sod shock tube with Roe solver, Superbee slope
    • 4.4.8 Example: Sod shock tube with Roe solver, PP reconstruction
  • 4.5 Alternative concepts
• 5 Applications
  • 5.1 Composing operators
    • 5.1.1 Operator adding versus splitting
    • 5.1.2 Godunov versus Strang operator splitting
    • 5.1.3 Steady-state solutions with operator adding or splitting
    • 5.1.4 Linear stability of operator adding or splitting
    • 5.1.5 Going multi-dimensional
  • 5.2 Coupling of hydrodynamics and radiation transport
    • 5.2.1 Coupling
    • 5.2.2 Stability requirements
    • 5.2.3 Approximations
  • 5.3 Improvements
    • 5.3.1 Non-Cartesian and/or refined grids
    • 5.3.2 Optimization strategies
• A. Nomenclature
  • A..1 Quantities
  • A..2 Vector notation
• B. Exercises
  • B..1 Linear advection
  • B..2 Expanding SN bubble hits molecular cloud
• C. Schedule
• D. References
  • D..1 Lecture Notes
  • D..2 Books
  • D..3 Articles
  • D..4 Hydrodynamic codes
• Index