CompactCourse: Introduction to parallel-in-time and other new time-stepping methods - Summer19

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Summer 19
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 13:00 - 14:30, room 02.07.023
All students interested in simulation of time depending problems, in

particular students of BGCE, TopMath, CSE, Mathematics, Informatics and Mechanical/Electrical Engineers

programming assignment and written short report
Semesterwochenstunden / ECTS Credits
1 credit



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.


  • Introduction to ODEs/PDEs and standard time integration methods

(Runge-Kutta, explicit/implicit, linear/non-linear, splitting methods)

  • Parareal
  • ML-SDC
  • 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)
- 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
  • 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
  • Spectral Deferred Correction (SDC) Methods (Day 3, Session 2)
  • Multi-level SDC + PFASST (Day 4, Session 1)
  • Lagrangian methods (Day 4, Session 2)