CFD Methods

• Spectral Methods
  • Examples of spectral methods
  • Example 1 of spectral methods
  • Example 2 of spectral methods
• Fourier Series
  • Wave mathematical representation
  • Orthogonal Function System and Series Expansions of Function
  • The Fourier series of the period \(T=2L\)
  • The Fourier series of the period \(T=2\pi\)
  • Fourier series of odd and even functions \(T=2\pi\)
  • Even and Odd Extensions
  • Period and Angular Frequency
  • Complex Form of Fourier Series
  • Applications of Fourier Series to Differential Equations
  • Discrete Fourier Transform (DFT) and its Relation to Fourier Series
  • Fourier Series
  • Definition of the Fourier Transform and its inverse
  • Integration by parts
  • Fourier Transforms of derivatives
  • The Fourier Transform of a second derivative
  • Fourier Transform solutions of PDEs
  • Fourier Sine and Cosine Transforms
  • Some Fourier Sine Transform examples.
  • The Discrete Fourier Transform
• Iterative Solver
  • Iterative Solver
  • Stationary Methods: Gauss-Seidel
  • Jacobi’s Method
  • Gauss-Seidel iterative method
  • Why the matrix-based formula works
  • Solving the 1D Poisson equation using finite differences
  • Solving the 2D Poisson equation using finite differences
  • Non-Stationary Methods: Conjugate Gradient Algorithm
  • Conjugate gradient algorithm
  • Linear Conjugate Gradient Algorithm
  • The Method of Steepest Descent
  • A-orthogonality
  • The conjugate gradient method
  • Example 1
  • Solving the 1D Poisson equation using conjugate gradient method
  • another form
  • The conjugate gradient method from code
  • Solving the 2D Poisson equation
• MULTIGRID
  • Multigrid Framework
  • Basic Iterative Methods
  • Residuals and Errors.
  • Model Problems
  • Gauss–Seidel method
  • Matrix Norms.
  • Interpreting the Spectral Radius.
  • Eigenvalues of the model problem
  • Numerical experiments
  • Gauss–Seidel iteration matrix
  • Gauss–Seidel eigenvalues and eigenvectors.
• MULTIGRID Continued
  • Elements of Multigrid
  • Two-Grid Correction Scheme
  • Numerical example.
  • V-Cycle Scheme
  • V-Cycle Scheme (Recursive Definition)
  • µ-Cycle Scheme
  • Full Multigrid V-Cycle (Recursive Form)
  • Full Multigrid V-Cycle
• Multigrid Poisson Solver
  • Problem Description
  • Discretization
  • Forming the Linear System
  • Classical Iterative Methods
  • Algorithm 1 Multigrid V-Cycle
  • Restriction Operator
  • Prolongation Operator
  • Restriction Operator Continue
  • Interpolation or Prolongation
• Vorticity Stream Function
  • Incompressible N-S Equation in 2D
  • Nondimensionalization
  • The Vorticity Equation
  • The Stream Function Equation
  • Boundary Conditions for the Stream Function
  • Discretizing the Domain
• Incompressible Navier-Stokes Equation
  • Incompressible N-S Equation in 2D
  • Lid-Driven Cavity Problem
  • Stream Function Equation
  • Boundary Conditions
  • Body Surface
  • Hybrid Arakawa-Spectral Solver