Introduction

• What is a PDE?
• What is a solution of PDE?
• Some commonly used PDE examples
• IVP and BVP
  
• Classification of PDEs
• Solution techniques 
• Analytic: PDE--> ODE
• Approximate semi-analytic; numerical solutions

Finite Difference Method for IVP

• Forward, Backward Euler's method, Crank-Nicholson
• Predictor-corrector method
  
• Runge-Kutta Method
• Matlab ODE Suite and usage

Finite Difference Method for two-point boundary value problems

• Finite difference discretization
• Dirichlet, Neumann, Robin boundary conditions and ghost point method
• Local truncation error, global error,  and convergence analysis
• Grid refinement analysis
• Non-linear problem and eigenvalue problems

Finite Difference Method for 2D Elliptic PDES

• Finite difference discretization and matrix vector form
• How to solve the system of equations
• Nine-point 4-th order compact scheme
• Maximum principle and error analysis
• finite difference scheme for polar coordinates

Finite Difference Method for parabolic PDEs

• Method of lines (MOL)
• Finite difference discretization (time and space)
• von-Neumann stability analysis and discrete Fourier transform

Finite Difference Method for Hyperbolic PDEs

• advection equation and correct boundary conditions
• CFL constarint
• Lax-Fridrichs, Lax-wendroff schemes
• Modified PDES and numerical scheme
• Conservation Laws and finite difference method

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