Computational methods
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
Discretization frameworks
- Finite difference, finite volume, and finite element paradigms
- Consistency, stability, and convergence for discretizations
- CFL conditions and von Neumann analysis (introduction)
- Adaptive discretization strategies
- Meshless and particle methods (overview)
Linear and nonlinear solvers
- Sparse matrix storage and direct solvers
- Krylov methods and preconditioning in practice
- Newton–Krylov methods for nonlinear systems
- Multigrid and domain decomposition (introduction)
- Parallel algorithms and scalability basics
Time integration and optimization
- Explicit and implicit time-stepping for ODE/PDE systems
- Stiff problems and A-stable methods
- Optimal control discretization (introduction)
- PDE-constrained optimization (overview)
- Uncertainty quantification via sampling and surrogate models
STEM / applied
Software engineering for scientific computing
- Version control, testing, and continuous integration for research code
- Profiling, memory management, and performance tuning
- GPU computing for linear algebra and PDEs (introduction)
- Workflow tools: containers, Make/SCons, and reproducible pipelines
- Visualization of large-scale simulation output
Project-based applications
- Team projects in CFD, structural mechanics, or imaging
- Coupling solvers in multiphysics settings
- Benchmarking against published test cases
- Documentation and presentation of computational studies
- Ethics and validation in computational science
Notes
Topics reflect common graduate computational methods syllabi at US universities. Courses are often project-driven and assume prior numerical analysis or FEM coursework.