CompactCourse: Introduction to parallel-in-time and other new time-stepping methods - Summer19
- Term
- Summer 19
- Lecturer
- Dr. Daniel Ruprecht, University of Leeds, Dr. Martin Schreiber, TUM; contact: Dr. rer. nat. Tobias Neckel
- Time and Place
- block course August 26-30, 2019, time: 9:00 - 10:30 and 12:00 - 13:30, room 02.07.023
- Audience
- All students interested in simulation of time depending problems, in
particular students of BGCE, TopMath, CSE, Mathematics, Informatics and Mechanical/Electrical Engineers
- Tutorials
- -
- Exam
- programming assignment and written short report
- Semesterwochenstunden / ECTS Credits
- 1 credit
- TUMonline
- n.a.
News
- July 29: The compact course is now open for registration. Please contact Daniel Ruprecht, Martin Schreiber, and Tobias Neckel in one mail directly.
Prerequisites
Although there will be a brief introduction to ordinary and partial differential equations and time integration methods, students are expected to be already familiar with them. A working knowledge of standard numerical methods such as finite differences, Runge-Kutta methods, etc. would be very helpful, but is not necessary. To fully benefit from the course, students will be given a range of programming assignments. Experience with Python will therefore be greatly beneficial.
Format of the Course
This is a compact course on time integration methods. A brief introduction to existing time integration methods as well as space discretization and PDE solvers will be given. Based on this, novel time integration methods such as parallel-in-time methods, exponential integration and variants of this will be discussed.
The course consists of 4 consecutive half days of lectures, discussions and assignments. Groups consisting of 3-4 students will implement programming assignments in Python.
Project report: Each individual participant has to hand in a short report of 1 to 2 pages on brief exercise tasks until Sunday, Sept. 1st, 12:00.
Content
- Introduction to ODEs/PDEs and standard time integration methods
(Runge-Kutta, explicit/implicit, linear/non-linear, splitting methods)
- Parareal
- ML-SDC
- PFASST
- Exponential Integration
- Semi-Lagrangian methods
Schedule & Content (preliminary)
- Time integration basics (Day 1, Session 1)
- HPC challenges
- ODEs and PDEs
- linear/non-linear ODEs + examples
- linear/non-linear PDEs + examples
- Runge-Kutta time integrators
- Explicit RK
- Butcher table
- Implicit RK
- Pade approximations
- Convergence, Consistency and Stability
- Stability function
- CFL condition
- Splitting methods
- Parareal (ODE) (Day 1, Session 2)
- Space discretization and PDE solvers (Day 2, Session 1)
- Space discretization (1D)
- Global basis functions
- Trigonometric (Fourier)
- High-order polynomials (Chebychev)
- Local basis functions
- Overlapping
- Nodal with local interpolation (FD)
- Superposition of local basis functions (FEM)
- Non-overlapping
- Continuous (SEM)
- Discontinuous (DG)
- Overlapping
- Global basis functions
- Space discretization (nD)
- Spherical harmonics
- Time integration of PDEs
- Global spectral
- Fourier
- Spherical Harmonics
- Finite differences
- Finite elements
- Classical finite elements
- Galerkin methods
- Spectral elements (continuous Galerkin methods)
- Discontinuous Galerkin methods
- (Dispersion errors)
- Theory
- Normal mode analysis
- Global spectral
- Space discretization (1D)
- Parareal (PDE) (Day 2, Session 2)
- Exponential integrator methods (Day 3, Session 1)
- Rational approximation of exponential integrators (REXI)
- Terry-REXI
- Cauchy-REXI
- Butcher-REXI
- ETDnRK methods
- Integrating factor method for direct solution
- Strang splitting
- REXI with spherical harmonics
- Rational approximation of exponential integrators (REXI)
- Spectral Deferred Correction (SDC) Methods (Day 3, Session 2)
- Multi-level SDC + PFASST (Day 4, Session 1)
- Lagrangian methods (Day 4, Session 2)
- Project work (Day 5, Sessions 1 + 2)