Numerical solution of partial differential equations
MTH 453/553

MWF 10:00-10:50AM
Zoom
Spring 2021

Outline

Last modified: Fri Mar 26 10:54:05 PDT 2021

Chapters and § refer to Finite Difference Methods for Ordinary and Partial Differential Equations by Randy LeVeque.

  1. Background on PDE Theory
    • § E.1: Examples and Classification
        Including
      • Elliptic, Parabolic, Hyperbolic
      • Steady-state, Dynamic
      • Linear, Non-linear
    • Cannonical Problems
    • § E.2: Derivation and Solutions
    • § E.3: Fourier Analysis and Well-posedness
  2. Chapter 2: Two-point BVP (ODE)
    • § 2.4: Simple Numerical Method
      • § 2.5-2.11: Stability, Consistency, Convergence
      • § 2.12: Neumann Boundary Conditions
    • Existence, Uniqueness, Well-posedness
    • Generalizations
  3. Chapter 3: Elliptic Equations
    • § 3.2: 5-point Stencil
    • § 3.4-3.5: Accuracy and Stability
  4. Chapter 4: Solution Methods for Resulting Linear Systems
    • § 4.1-4.2: Jacobi, Gauss-Seidel and SOR
    • § 4.3: Conjugate Gradients
    • § 4.4: GMRES
  5. Chapter 9: Parabolic Equations
    • Methods and Stencils
    • § 9.1: Accuracy
    • § 9.2-9.3: Stability
    • § 9.4: Stiffness
    • § 9.5: Convergence
    • § 9.6: Von Neumann Stability Analysis
    • Dissipation and Dispersion
    • Generalizations
  6. Chapter 10: Hyperbolic Equations
    • § 10.1: Advection
    • § 10.2: Stability
    • § 10.3: Higher Order Methods
    • § 10.4: Upwind Methods
    • § 10.5: Von Neumann Stability Analysis
    • § 10.7: CFL Condition
    • Dissipation and Dispersion
    • Generalizations
    • Wave Equation