# Numerical methods

### From CFD-Wiki

(Difference between revisions)

(48 intermediate revisions not shown) | |||

Line 2: | Line 2: | ||

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]] |

- | + | #### [[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 Poisson's equation]] | ||

- | # [[Solution of Navier-Stokes | + | # [[Solution of Navier-Stokes equations]] |

## [[Velocity-pressure coupling]] | ## [[Velocity-pressure coupling]] | ||

### [[Rhie-Chow interpolation]] | ### [[Rhie-Chow interpolation]] | ||

Line 62: | Line 52: | ||

### [[PWIM - Pressure Weighted Interpolation Method]] | ### [[PWIM - Pressure Weighted Interpolation Method]] | ||

### [[ACM - Artificial Compressibility 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 | ||

+ | ## [[Area calculations]] | ||

+ | ## [[Volume calculations]] | ||

+ | # Discretization | ||

+ | ## [[Finite difference]] | ||

+ | ## [[Finite volume]] | ||

+ | ### [[Generic scalar transport equation]] | ||

+ | ### [[Discretization of the diffusion term]] | ||

+ | ### [[Discretization of the convection term]] | ||

+ | ### [[Discretization of the transient term]] | ||

+ | ### [[Source term linearization]] | ||

+ | ### [[Gradient computation]] | ||

+ | ## [[Spectral volume]] | ||

+ | ##[[Discontinuous Galerkin]] | ||

+ | ## [[Finite element]] | ||

+ | ## [[Time discretisation]] | ||

+ | ### [[Euler method]] | ||

+ | ### [[Predictor corrector methods]] | ||

+ | #### [[Runge Kutta methods]] | ||

+ | #### [[Adams Bashforth method]] | ||

+ | #### [[Adams Moulton method]] | ||

+ | ### [[Implicit second order method]] | ||

+ | # Linear systems of equations | ||

+ | ## [[Introduction and need]] | ||

+ | ## [[Direct methods]] | ||

+ | ### [[Gaussian elimination]] | ||

+ | ### [[LU decomposition]] | ||

+ | ### [[Tridiagonal matrix algorithm - TDMA (Thomas algorithm)]] | ||

+ | ## [[Iterative methods]] | ||

+ | ### [[Gauss-Seidel method]] | ||

+ | ### [[Jacobi method]] | ||

+ | ### [[Successive over-relaxation method - SOR]] | ||

+ | ### [[Stone's method]] | ||

+ | ### [[Alternating direction implicit (ADI) method]] | ||

+ | ### [[Conjugate gradient methods]] | ||

+ | #### [[Conjugate gradient method of Golub and van Loan]] | ||

+ | #### [[Biconjugate gradient method]] | ||

+ | #### [[Biconjugate gradient stabilized method]] | ||

+ | ## Matrix factorization and preconditioning | ||

+ | ### [[Incomplete LU factorization - ILU]] | ||

+ | ### [[Incomplete Cholesky factorization]] | ||

+ | ## [[Multigrid methods]] | ||

+ | ### [[Geometric multigrid - FAS]] | ||

+ | ### [[Algebraic multigrid - AMG]] | ||

# [[Efficiency and stability]] | # [[Efficiency and stability]] | ||

## Limiters | ## Limiters | ||

### [[Flux limiters]] | ### [[Flux limiters]] | ||

### [[Gradient limiters]] | ### [[Gradient limiters]] | ||

- | |||

# [[Meshless methods]] | # [[Meshless methods]] | ||

# [[Overset grids]] | # [[Overset grids]] | ||

# [[Panel method]] | # [[Panel method]] | ||

- | + | # [[Calculation on non-orthogonal curvelinear structured grids, finite-volume method]] | |

- | + | ||

- | + | ||

- | + |

## Latest revision as of 12:42, 26 August 2012

## Numerical Aspects of CFD

This section covers the numerical soul of CFD.

- Introduction to numerical methods
- Basic aspects of discretization
- Solution of Euler equation
- Solution of Poisson's equation
- Solution of Navier-Stokes equations
- Velocity-pressure coupling
- 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
- CIP - Cubic Interpolating Polynomial Method

- Discrete Operator Splitting

- Solution of Boltzmann Equation
- CFD Related algorithms
- Wall distance Calculations

- Geometrical Calculations
- Discretization
- Linear systems of equations
- Introduction and need
- Direct methods
- Iterative methods
- Matrix factorization and preconditioning
- Multigrid methods

- Efficiency and stability
- Limiters

- Meshless methods
- Overset grids
- Panel method
- Calculation on non-orthogonal curvelinear structured grids, finite-volume method