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)