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Numerical methods

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

Latest revision as of 12:42, 26 August 2012

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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