### 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

#### 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.

#### 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

##### 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