Difference between revisions of "CompactCourse: Introduction to parallel-in-time and other new time-stepping methods - Summer19"

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| term = Summer 19
 
| term = Summer 19
 
| lecturer = [https://engineering.leeds.ac.uk/staff/741/dr_daniel_ruprecht Dr. Daniel Ruprecht, University of Leeds], [https://www.caps.in.tum.de/en/staff/martin-schreiber/ Dr. Martin Schreiber, TUM]; contact: [[Dr. rer. nat. Tobias Neckel]]
 
| lecturer = [https://engineering.leeds.ac.uk/staff/741/dr_daniel_ruprecht Dr. Daniel Ruprecht, University of Leeds], [https://www.caps.in.tum.de/en/staff/martin-schreiber/ Dr. Martin Schreiber, TUM]; contact: [[Dr. rer. nat. Tobias Neckel]]
| timeplace = block course August 26-30, 2019, time: 9:00 - 10:30 and 13:00 - 14:30, room 02.07.023
+
| timeplace = block course August 26-30, 2019, time: 9:00 - 10:30 and 12:00 - 13:30, room 02.07.023
 
| credits = 1 credit
 
| credits = 1 credit
 
| audience =  
 
| audience =  
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*** linear/non-linear ODEs + examples
 
*** linear/non-linear ODEs + examples
 
*** linear/non-linear PDEs + examples
 
*** linear/non-linear PDEs + examples
- Runge-Kutta time integrators
+
** Runge-Kutta time integrators
  + Explicit RK
+
*** Explicit RK
  + Butcher table
+
*** Butcher table
  + Implicit RK
+
*** Implicit RK
  + Pade approximations
+
*** Pade approximations
- Convergence, Consistency and Stability
+
** Convergence, Consistency and Stability
- Stability function
+
** Stability function
- CFL condition
+
** CFL condition
- Splitting methods
+
** Splitting methods
  
 
* Parareal (ODE) (Day 1, Session 2)
 
* Parareal (ODE) (Day 1, Session 2)
  
 
* Space discretization and PDE solvers (Day 2, Session 1)
 
* Space discretization and PDE solvers (Day 2, Session 1)
- Space discretization (1D)
+
** Space discretization (1D)
  + Global basis functions
+
*** Global basis functions
    - Trigonometric (Fourier)
+
**** Trigonometric (Fourier)
    - High-order polynomials (Chebychev)
+
**** High-order polynomials (Chebychev)
  + Local basis functions
+
*** Local basis functions
    - Overlapping
+
**** Overlapping
      + Nodal with local interpolation (FD)
+
***** Nodal with local interpolation (FD)
      + Superposition of local basis functions (FEM)
+
***** Superposition of local basis functions (FEM)
    - Non-overlapping
+
**** Non-overlapping
      + Continuous (SEM)
+
***** Continuous (SEM)
      + Discontinuous (DG)
+
***** Discontinuous (DG)
 
+
** Space discretization (nD)
- Space discretization (nD)
+
*** Spherical harmonics
    + Spherical harmonics
+
** Time integration of PDEs
 
+
*** Global spectral
- Time integration of PDEs
+
**** Fourier
  + Global spectral
+
**** Spherical Harmonics
    - Fourier
+
*** Finite differences
    - Spherical Harmonics
+
*** Finite elements
  + Finite differences
+
**** Classical finite elements
  + Finite elements
+
**** Galerkin methods
    - Classical finite elements
+
***** Spectral elements (continuous Galerkin methods)
    - Galerkin methods
+
***** Discontinuous Galerkin methods
      + Spectral elements (continuous Galerkin methods)
+
*** (Dispersion errors)
      + Discontinuous Galerkin methods
+
**** Theory
 
+
**** Normal mode analysis
  + (Dispersion errors)
 
    - Theory
 
    - Normal mode analysis
 
 
 
 
* Parareal (PDE) (Day 2, Session 2)
 
* Parareal (PDE) (Day 2, Session 2)
 
 
* Exponential integrator methods (Day 3, Session 1)
 
* Exponential integrator methods (Day 3, Session 1)
  - Rational approximation of exponential integrators (REXI)
+
** Rational approximation of exponential integrators (REXI)
    + Terry-REXI
+
*** Terry-REXI
    + Cauchy-REXI
+
*** Cauchy-REXI
    + Butcher-REXI
+
*** Butcher-REXI
  - ETDnRK methods
+
** ETDnRK methods
  - Integrating factor method for direct solution
+
** Integrating factor method for direct solution
  - Strang splitting
+
** Strang splitting
  - REXI with spherical harmonics
+
** REXI with spherical harmonics
 
 
 
* Spectral Deferred Correction (SDC) Methods (Day 3, Session 2)
 
* Spectral Deferred Correction (SDC) Methods (Day 3, Session 2)
 
 
* Multi-level SDC + PFASST (Day 4, Session 1)
 
* Multi-level SDC + PFASST (Day 4, Session 1)
 +
* Lagrangian methods (Day 4, Session 2)
  
* Lagrangian methods (Day 4, Session 2)
+
* Project work (Day 5, Sessions 1 + 2)
  
== Material ==
 
t.b.a.
 
  
  
 
[[Category:Teaching]]
 
[[Category:Teaching]]

Latest revision as of 09:37, 27 November 2019

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

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)