Problems Leading to PDEs. Classification of Second Order PDEs

Finite Difference Approximation

Derivation of Finite Difference Approximation

Basic Concepts: Consistency, Stability, and Convergence

Neumann Boundary Conditions.

Existence and Uniqueness

Stability using L2 norm

General Linear Second Order Equations

Singular Perturbation Problems and Nonuniform Grids

Spectral Method

Finite Element Method

Finite Element Method. Nonhomogeneous BCs

Numerical Solution for Elliptic PDEs. Laplace Equation

The 9-Point Stencil for Laplace and Poisson. Equation Based FDM

Accuracy and Stability of Elliptic Equations

Deferred Correction Poisson Equation in 1D

Deferred Correction for Laplace and Poisson Equations

Acoustic Scattering and Fafield Expansiosn ABCs

Diffusion Equations and Parabolic Problems

Crank Nicholson and Weighted Average Scheme for the Heat Equation

Stability Time-Dependent Problems. Maximum Principle

Matrix Stability. Lax Equivalence Theorem

Matrix Stability Analysis Applied to Weighted Avergae Scheme for the Heat Equation with Neumann BC

Multi-Dimensional Parabolic Problems and Crank-Nicholson

Alternating Direction Implicit finite difference method for the Heat Equation

Tridiagonal Algorithm. Crout Factorization

Matrix Norms

Iterative Methods for Linear Systems (Brief))

Iterative Methods for Linear Systems (Full)

A 3x3 example of Iterative Methods for Linear Systems

Convergence Theorems for Iterative Methods for Linear Systems

SOR applied to Poisson's Equation'

SOR applied to Poisson's Equation' Short Version

Elliptic Grid Generation. Amsden-Hirt

Winslow Grid Generation

Grid Quality