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

Line 57: | Line 57: | ||

*** linear/non-linear ODEs + examples | *** linear/non-linear ODEs + examples | ||

*** linear/non-linear PDEs + 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) | * 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) | |

− | + | *** 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) | * 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) | |

− | + | *** 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) | * 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) | ||

− | |||

== Material == | == Material == |

## Revision as of 12:59, 13 August 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 13:00 - 14:30, 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.

## Contents

# 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)

## Material

t.b.a.