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.

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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)
  • 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)