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 12:00 - 13:30, 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)
  • Project work (Day 5, Sessions 1 + 2)