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. Finite element
   4. Time discretisation
5. Linear equation systems
   1. Introduction and need
   2. Direct numerical solutions
      1. Gauss elimination
      2. LU decomposition method
      3. LU factorisation
         1. LU factorisation for sparse matrices
      4. Direct solution to tridiagonal matrix
         1. Thomas algorithm
   3. 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
   4. Multigrid methods
      1. Geometric multigrid
      2. Additive corrective multigrid
   5. Solution of Poisson's equation
   6. Solution of Navier-Stokes equation
6. Efficiency and stability
   1. Limiters
      1. Flux limiters
      2. Gradient limiters