Maria K. Cameron

AMSC/CMSC 661: Scientific Computing II

The curriculum was updated in Spring 2023

Ordinary differential equations

• Consistensy, Stability, Convergence
• Linear Stability Theory
• Runge-Kutta Methods
• Multistep Methods (Adams, BDF)
• Symplectic Methods for Integrating Hamiltonian systems

Lecture notes: ODEsolvers.pdf, SymplecticMethods.pdf

Numerical methods for elliptic PDEs

• Linear elliptic equations. Modeling using elliptic PDEs. Existence and Uniqueness theorems, weak and strong maximum principles.
• Finite difference methods in 2D: different types of boundary conditions, convergence.
• Variational and weak formulations for elliptic PDEs.
• Finite element method in 2D. Implementation. Convergence analysis.
• Neural network-based solvers.

Lecture notes: elliptic.pdf, NeuralNetworks4PDEs.pdf

Numerical Methods for Time-Dependent PDEs

• Parabolic equations:
  • Heat equation. Finite difference methods: explicit and implicit. Basic facts about stability and convergence.
  • Solving heat equation in 2D using finite element method.
  • Method of lines.
  
  
• Linear advection equation:
  • Finite difference methods. Basic facts about stability and convergence. The CFL condition.
  • Fourier transform. Dispersion analysis. Phase and group velocities.


• Hyperbolic conservation laws:
  • Shock speed and the Rankine-Hugoriot condition, weak solutions, entropy condition and vanishing viscosity solution
  • Numerical methods for conservation laws: conservative form, consistency, Godunov's method.
• Spectral methods for solving linear and nonlinear PDEs
• Linear dispersive equations. Korteweg-de Vries equation. Kuramoto-Sivashinski equation

Lecture notes: Hyperbolic.pdf, spectral.pdf

HW collection from Spring 2023 (HW 1 - 13)

Take-home final exam: Spring 2023

Some references

• [1] Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 9, Chapter 10, Appendix E


• [2] Jochen Alberty, Carsten Carstensen and Stefan A. Funken, "Remarks around 50 lines of Matlab: short finite element implementation", Numerical Algorithms 20 (1999) 117–137


• [3] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Second Edition, Cambridge University Press, 2005


• [4] S. Larsson and V. Thomee, Partial Differential Equations with Numerical Methods


• [5] John P. Boyd, Chebyshev and Fourier Spectral methods, 2nd edition, Dover Publication, Inc., Mineola, New York, 2001