# Maria K. Cameron

### University of Maryland, Department of Mathematics

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### AMSC/CMSC 661: Scientific Computing II

##### 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

##### Numerical Linear Algebra for Sparse Matrices
• Basic iterative methods: Jacobi, Gauss-Seidel, SOR.

• Multigrid.

• Krylov subspace methods: the conjugate gradient and generalizations
##### 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. An example of a nonlinear equation (the Boussinesq 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 and Glimm's methods.
##### Fourier and Wavelet Transform Methods
• Continuous and Discrete Fourier transforms

• Spectral methods for solving linear and nonlinear PDEs

• The fast Fourier transform

• Nyquist frequency, sampling theorem

• Continuous and discrete wavelet transforms

• Haar and Daubechies wavelets, approximation properties, fast wavelet transforms

• Application of wavelets to image processing