Numerical methods

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Numerical Aspects of CFD

This section covers the numerical soul of CFD.

1. Introduction to numerical methods
2. Geometrical Calculations
   1. Area calculations
   2. Volume calculations
3. Gradient calculations
4. Discretisation
   1. Finite differences
   2. Finite volume
      1. Discretisation of convective term
      2. Discretisation of diffusive term
      3. Discretisation on orthogonal grids
      4. Discretisation on non-orthogonal curvilinear body-fitted grids
      5. I do not think we need 3 and 4, since 1 and 2 deals with non ortho and ortho both!!!!
   3. Finite element
   4. Time discretisation
5. Linear equation systems
   1. Introduction and need
   2. Matrix Related
      1. Matrix factorisation
         1. LU factorisation for sparse matrices
         2. Incomplete Cholesky factorization
   3. Direct numerical solutions
      1. Gauss elimination
      2. LU decomposition method
      3. Direct solution to tridiagonal matrix
         1. Thomas algorithm
   4. Iterative solutions
      1. Basic concept of iterative solutions
      2. Gauss-Seidel method
      3. Jacobi method
      4. Successive over-relaxation method
      5. Stone's method
      6. Alternating direction implicit (ADI) method
      7. Congugate gradient methods
         1. Conjugate gradient method of Golub and van Loan
         2. Biconjugate gradient method
   5. Multigrid methods
      1. Geometric multigrid
      2. Additive corrective multigrid
   6. Solution of Poisson's equation
   7. Solution of Navier-Stokes equation
      1. SIMPLE algorithm
      2. SIMPLEC algorithm
      3. SIMPLER algorithm
      4. PISO algorithm
6. Efficiency and stability
   1. Limiters
      1. Flux limiters
      2. Gradient limiters
7. Meshless methods
8. CFD Related algorithms
   1. Wall distance Calculations
      1. Moore's kd-tree algorithm
      2. Transport equation based wall distance calculation