Post on 31-Aug-2018
Finite Differences - Basics
FINITE DIFFERENCES:Taylor Series, Higher Order Accuracy
FINITE DIFFERENCES:Taylor Series, Higher Order Accuracy Cont’d
Forward Differences: Forward finite-divided-difference formulas:
Backward Differences: Backward finite-divided-difference formulas:
Centered Differences:Centered finite-divided-difference formulas:
FINITE DIFFERENCES: Taylor Series, Higher Order Accuracy
FINITE DIFFERENCES:Interpolation Formulas for Higher Order Accuracy
Error Types and Discretization Properties:Consistency
Error Types and Discretization Properties:Truncation error and Error equation
Error Types and Discretization Properties:Stability
Error Types and Discretization Properties:Convergence
Review
Review (2)
FINITE DIFFERENCES: Higher Order Accuracy:Taylor Tables
FINITE DIFFERENCESHigher Order Accuracy: Taylor Tables Cont’d
FINITE DIFFERENCESHigher Order Accuracy: Taylor Tables Cont’d
Higher-Order Finite Difference SchemesConsiderations
Finite Difference Schemes:Implementation of Boundary conditions
Finite Difference Schemes:Implementation of Boundary conditions, Cont’d
Finite-Differences on Non-Uniform Grids: 1-D
Non-Uniform Grids Example: 1-D Central-difference
Non-Uniform Grids Example: 1-D Central-difference
Non-Uniform Grids Example: 1-D Central-differenceConclusions
Grid-Refinement and Error estimation
Richardson Extrapolation and Romberg Integration
Fourier (Error) Analysis:Definitions
Fourier (Error) Analysis:Differentiations
Fourier (Error) Analysis:Generic equation
Fourier (Error) Analysis:Generic equation
Fourier Error Analysis: 1st derivatives
Fourier Error Analysis, Cont’d:Effective Wave numbers
Fourier Error Analysis, Cont’dEffective Wave Speeds
Evaluation of the Stability of a FD Scheme
Von Neumann Stability
Evaluation of the Stability of a FD SchemeVon Neumann Example
Evaluation of the Stability of a FD SchemeVon Neumann Example
Evaluation of the Stability of a FD Schemevon Neumann Example
Partial Differential EquationsHyperbolic PDE: B2 -4 A C > 0
Partial Differential EquationsHyperbolic PDE
Partial Differential EquationsHyperbolic PDE
Partial Differential EquationsHyperbolic PDE
Courant-Fredrichs-Lewy Condition (1920’s)
CFL: Linear convection (Sommerfeld Eqn) Example
CFL: 2nd order Wave equation Example
CFL Condition: Some comments
von Neumann Examples
Partial Differential EquationsElliptic PDE
Partial Differential EquationsElliptic PDEs
Partial Differential EquationsElliptic PDE
Elliptic PDEsLaplace Equation, Global Solvers
Ellipticic PDEsNeumann Boundary Conditions
Elliptic PDEs
Elliptic PDE:Poisson Equation
Elliptic PDE:Poisson Equation
Helmholtz Equation
Elliptic PDE’sHigher Order Finite Differences
Elliptic PDEs: Irregular Boundaries
Elliptic PDEs: Irregular Boundaries
Elliptic PDEsInternal Boundaries
Elliptic PDEsInternal Boundaries –Higher Order
Partial Differential Equations Parabolic PDE:B2 -4 A C = 0
Partial Differential EquationsParabolic PDE: 1D Heat Flow example
Parabolic PDE1D Heat Flow: Forward in time, centered in space, explicit
1D Heat Flow: Forward in time, centered in space, explicit
Leads to a system of equations to be solved at each time-step
Parabolic PDE: Implicit SchemesCrank-Nicolson Scheme
Parabolic PDEs: Implicit SchemesCrank-Nicolson – r = 1
Parabolic PDEs: Two spatial dimensions
Parabolic PDEs: Two spatial dimensions
Parabolic PDEs: Two spatial dimensionsADI scheme (Two Half steps in time)
Parabolic PDEs: Two spatial dimensionsADI scheme (Two Half steps in time)