[1] General Introduction to PDEs

• Classification of first order PDEs
  Linear vs. Quasilinear vs. Nonlinear
• Classification of second order PDEs
  Linear vs. Quasilinear vs. Nonlinear
  Elliptic vs. Parabolic vs. Hyperbolic
  Time dependent vs. independent
  Steady State vs. No Steady State
• Example PDEs:
  Advection Equation
  Laplace Equation
  Heat Equation
  Wave Equation
• Definitions:
  Well Posed
  Characteristic
  Domain of Dependence
  Range of Influence
• How to Compute:
  Characteristics of a first-order PDE
  Characteristics for the wave equation

[2] Parabolic in 1D

• Heat Equation:
  Exact solution
• Finite Differences:
  Forward vs. Backward vs. Centered
  Accuracy using Taylor Series
• Numerical Methods:
  Explicit Method
  Implicit Method
  Theta Method
  Crank Nicolson
  Method of Lines
• Error and Stability Analysis:
  Truncation Error
  Fourier Analysis
  Maximum Principle
  Global error vs. Truncation Error
• Boundary Conditions:
  Dirichlet
  Neumann
  Robin
• More general Parabolic problems:
  Setup explicit/implicit methods
  Cost of each timestep
  Upwind scheme
• How to compute:
  Stability using Fourier
  Convergence using maximum principle
  One step with an explicit method

[3] Parabolic in higher dimensions

• Stability of Methods:
  Explicit
  Crank Nicolson
  ADI
• Accuracy of Methods:
  Explicit
  Crank Nicolson
  ADI
• Cost of Methods:
  Explicit
  Crank Nicolson
  ADI
• More general boundaries:
  Dirichlet on a curved boundary
• How to Compute:
  Stability using Fourier
  One step with an explicit method
  Discrete equations at curved boundary

[4] Hyperbolic Problems

• Stability:
  Characteristics
  Courant-Friedrich-Lewy (CFL)
  Domain of Dependence
• Fourier Analysis:
  Upwind Method
  Centered Difference
• Existence and Uniqueness
  Crossing of Characteristics
  Conservation form
  Weak solution in conservation form
  Shock speed from conservation form
• Numerical method properties:
  Upwind Method, Lax-Wendroff
• Know how to compute:
  One step with an explicit method
  Stability using Fourier
  Crossing time of characteristics
  Conservation form
  Shock speed

[6] Elliptic Problems

• Discretization:
  Centered difference scheme
  Sparse structure of equations
  Local order of accuracy
• Global convergence:
  Comparison function
  Conditions for Theorem 6.1
  Result for curved boundaries
  Result for Neumann boundary conditions