Computational methods
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Computational Methods — scope drawn from open computational mathematics notes and typical US graduate computational methods syllabi.
Discretization frameworks
- Finite difference and finite volume methods — Finite difference and finite volume methods
- Finite element paradigms — Finite element paradigms
- Consistency and stability — Consistency and stability
- Convergence for discretizations — Convergence for discretizations
- CFL conditions — CFL conditions
- Von Neumann analysis (introduction) — Von Neumann analysis (introduction)
- Adaptive discretization strategies — Adaptive discretization strategies
- Meshless and particle methods (overview) — Meshless and particle methods (overview)
Linear and nonlinear solvers
- Sparse matrix storage and direct solvers — Sparse matrix storage and direct solvers
- Krylov methods — Krylov methods
- Preconditioning in practice — Preconditioning in practice
- Newton–Krylov methods for nonlinear systems — Newton–Krylov methods for nonlinear systems
- Multigrid — Multigrid
- Domain decomposition (introduction) — Domain decomposition (introduction)
- Parallel algorithms — Parallel algorithms
- Scalability basics — Scalability basics
Time integration and optimization
- Explicit and implicit time-stepping for ODE/PDE systems — Explicit and implicit time-stepping for ODE/PDE systems
- Stiff problems and A-stable methods — Stiff problems and A-stable methods
- Optimal control discretization (introduction) — Optimal control discretization (introduction)
- PDE-constrained optimization (overview) — PDE-constrained optimization (overview)
- Uncertainty quantification via sampling — Uncertainty quantification via sampling
- Surrogate models — Surrogate models
Software engineering for scientific computing
- Version control and testing — Version control and testing
- Continuous integration for research code — Continuous integration for research code
- Profiling and memory management — Profiling and memory management
- Performance tuning — Performance tuning
- GPU computing for linear algebra — GPU computing for linear algebra
- PDEs (introduction) — PDEs (introduction)
- Workflow tools — Workflow tools
- Reproducible pipelines — Reproducible pipelines
- Visualization of large-scale simulation output — Visualization of large-scale simulation output
Project-based applications
- Team projects in CFD, structural mechanics, or imaging — Team projects in CFD, structural mechanics, or imaging
- Coupling solvers in multiphysics settings — Coupling solvers in multiphysics settings
- Benchmarking against published test cases — Benchmarking against published test cases
- Documentation — Documentation
- Presentation of computational studies — Presentation of computational studies
- Ethics and validation in computational science — Ethics and validation in computational science
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Discretization frameworks
Finite difference and finite volume methods
Coming soon - Linear and nonlinear solvers
Sparse matrix storage and direct solvers
Coming soon - Time integration and optimization
Explicit and implicit time-stepping for ODE/PDE systems
Coming soon - Software engineering for scientific computing
Version control and testing
Coming soon - Project-based applications
Team projects in CFD, structural mechanics, or imaging
Coming soon
Notes
Graduate Computational Methods — scope drawn from open computational mathematics notes and typical US graduate computational methods syllabi. Topic outline: `content/topics/graduate/computational_methods.json`.