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Numerical methods

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Numerical Aspects of CFD

This section covers the numerical soul of CFD.

  1. Introduction to numerical methods
  2. Basic aspects of discretization
    1. Classification of govering equations
      1. Integral form of govering equations
      2. Differential form of govering equations
      3. Forms of the govering equations suited for CFD
      4. Generic scalar transport equation
    2. Mathematical behaivior of Partial Differential Equations
      1. Classification of physical behaviour
        1. Equilibrium problems
        2. Marching problems
      2. Mathematical Classification of flows
        1. Hyperbolic flows
        2. Parabolic flows
        3. Elliptic flows
        4. Mixed flow types
      3. Stability Analysis
        1. Fourier stability analysis
        2. von Neuman stability analysis
        3. Courant–Friedrichs–Lewy condition
    3. Properties of Numerical Solution Methods
      1. Consistency
      2. Stability
      3. Conservation
      4. Boundedness
      5. Realizability
      6. Accuracy
  3. Solution of Euler equation
  4. Solution of Poisson's equation
  5. Solution of Navier-Stokes equations
    1. Velocity-pressure coupling
      1. Rhie-Chow interpolation
    2. Fully coupled methods - FC
    3. DeCoupled methods - DC
      1. SIMPLE algorithm
      2. SIMPLEC algorithm - SIMPLE Consistent
      3. SIMPLEM algorithm - SIMPLE-Modified
      4. SIMPLEX algorithm
      5. SIMPLEST algorithm - SIMPLE ShorTened
      6. SIMPLER algorithm - SIMPLE - Revised
      7. PISO algorithm - Pressure Implicit with Split Operator
      8. PRIME algorithm - PRessure Implicit Momentum Explicit
      9. MSIMPLEC, MPISO , SIMPLESSEC , SIMPLESSE
      10. CPC - Compressible Pressure Correction algorithm
      11. MCBA - Mass Conservation Based Algorithms
      12. GCBA - Geometric Conservation Based Algorithms
      13. IPSA - Inter-Phase Slip Algorithm
      14. CPI - Consistent Physical Interpolation
      15. MWIM - Momentum Weighted Interpolation Method
      16. PWIM - Pressure Weighted Interpolation Method
      17. ACM - Artificial Compressibility Method
      18. CIP - Cubic Interpolating Polynomial Method
    4. Discrete Operator Splitting
  6. Solution of Boltzmann Equation
  7. CFD Related algorithms
    1. Wall distance Calculations
      1. Moore's kd-tree algorithm
      2. Transport equation based wall distance calculation
  8. Geometrical Calculations
    1. Area calculations
    2. Volume calculations
  9. Discretization
    1. Finite difference
    2. Finite volume
      1. Generic scalar transport equation
      2. Discretization of the diffusion term
      3. Discretization of the convection term
      4. Discretization of the transient term
      5. Source term linearization
      6. Gradient computation
    3. Spectral volume
    4. Discontinuous Galerkin
    5. Finite element
    6. Time discretisation
      1. Euler method
      2. Predictor corrector methods
        1. Runge Kutta methods
        2. Adams Bashforth method
        3. Adams Moulton method
      3. Implicit second order method
  10. Linear systems of equations
    1. Introduction and need
    2. Direct methods
      1. Gaussian elimination
      2. LU decomposition
      3. Tridiagonal matrix algorithm - TDMA (Thomas algorithm)
    3. Iterative methods
      1. Gauss-Seidel method
      2. Jacobi method
      3. Successive over-relaxation method - SOR
      4. Stone's method
      5. Alternating direction implicit (ADI) method
      6. Conjugate gradient methods
        1. Conjugate gradient method of Golub and van Loan
        2. Biconjugate gradient method
        3. Biconjugate gradient stabilized method
    4. Matrix factorization and preconditioning
      1. Incomplete LU factorization - ILU
      2. Incomplete Cholesky factorization
    5. Multigrid methods
      1. Geometric multigrid - FAS
      2. Algebraic multigrid - AMG
  11. Efficiency and stability
    1. Limiters
      1. Flux limiters
      2. Gradient limiters
  12. Meshless methods
  13. Overset grids
  14. Panel method
  15. Calculation on non-orthogonal curvelinear structured grids, finite-volume method
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