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Volume I / Governing Equations

  1. Part I Scalar Equations
    1. Scalar Equations
      1. Linear Advection Equations
      2. Burgers' Equation and KdV Equation
      3. Advection-Diffusion Equation
      4. Advection-Reaction Equation
      5. Reaction-Diffusion Equation
      6. Poisson Equation
  2. Part II Hyperbolic System
    1. Characteristic Analysis and Roe Matrix
      1. Identities
      2. Polytropic Gas Equations
        1. 1D
        2. 2D
        3. 3D
      3. Shallow water
        1. 1D
        2. 2D
      4. Euler Equations
        1. 1D
        2. 2D
        3. 3D
      5. MHD Equations
        1. 1D Polytrope MHD
        2. 1.5D and 2D Adiabatic MHD
        3. 1.75D, 2.5D and 3D Adiavatic MHD
          1. Linear Eigensystem
          2. Renormalized Eigensystem
          3. A Roe matrix
          4. Other Roe matrix
        4. 3D Isothermal MHD
      6. Ten Moment
      7. Lagrangian fluid
  3. Part III Hyperbolic System with Source Terms
    1. Stiff relaxation
    2. Geometrical Source Terms and Tips for Fininte Volume Discretization
      1. 1D Cylindrical Coordinate
      2. 1D Spherical Coordinate
      3. 2D Isothermal Disk
        1. Angular momentum conservation form
    3. Navier-Stokes Equations
      1. 3D
        1. Reynolds Number
        2. Prandtl Number
        3. Dissipation Function
      2. 2D
      3. 1D
    4. Resistive MHD
      1. Magnetic Relnolds Number
      2. Eight wave formalism

Volume II / Method of Discretization

  1. Finite Volume Method
    1. Numerical Flux
  2. Finite Difference Method
  3. Finite Element Method
    1. Discontinuous Galerkin
  4. Spectral Difference Method
  5. Spectral Volume Method

Volume III / Numerical Flux and Riemann Problem

  1. Lax-Friedrich/Local Lax-Friedlich
  2. HLL
  3. ASUM
  4. Roe
    1. Nature of Roe average
    2. Property U
    3. Entropy fix
  5. Exact Riemann Solver
    1. Two Shock Approximation

Volume IV / Spatial High Order Scheme

  1. MUSCL-type scheme
  2. van Leer's kappa Scheme
  3. PPM
  4. ENO/WENO
  5. RKDG
  6. CIP

Volume V / Order of Acuracy, Dissipation and Tests

  1. Operator Splitting
    1. Godunov Spritting
    2. Strang Splitting
  2. Accessories
    1. CFL number
    2. Artificial Viscosity
    3. Entropy fix
    4. von Neumann Stability Analysis
    5. Error Analysis
    6. TVD Limiter
      1. minmod
      2. double minmod
      3. van Leer limiter
      4. Superbee limiter
      5. van Albada limiter
      6. van Albada2
    7. TVB Limiter
  3. Test Problems
    1. Sod's Shock Tube
    2. Two Interacting Blastwaves
    3. Einfeldt Strong Rarefaction Test
    4. Shu and Osher Shocktube
    5. Entropy wave
    6. Shock Diffraction
    7. Double Mach Reflection
    8. Brio and Wu (1984)
    9. Torsional alfven wave
    10. Orzag-Tang Vortex
    11. Carbuncle

Volume VI / Techniques to get Steady Solutions

  1. Three time stepping
  2. Preconditioning
  3. Multigrid Iteration
  4. Rung-Kutta for High-frequency Dumping
  5. Dual timestepping

Volume VII / Beyond the Divergence Free Method

  1. Eight wave
  2. Projection
  3. Constrait Transport (CT)
  4. Locally divergence free basis function for DG

Volume VIII / Runge-Kutta Time Integration

  1. TVD Runge-Kutta
  2. High-frequenvy Dumping

Volume IX / Exact Solutions and Extrapolations for Accuracy Study

  1. Linear Advection
  2. Laplace Equation
  3. Cauchy-Riemann Equations (Potential Flows)
  4. Simple Waves
  5. Euler Equations
  6. Richardson's Extrapolation

uncategolyzed topics

  1. SPH
  2. coding tips
    1. fortran90
    2. MPI