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

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| 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 13:00 - 14:30, room 02.07.023 | ||

| credits = 1 credit | | credits = 1 credit | ||

− | | audience = | + | | audience = |

+ | All students interested in simulation of time depending problems, in | ||

+ | particular students of BGCE, TopMath, CSE, Mathematics, Informatics and | ||

+ | Mechanical/Electrical Engineers | ||

| tutorials = - | | tutorials = - | ||

| exam = programming assignment and written short report | | exam = programming assignment and written short report | ||

Line 12: | Line 15: | ||

= News = | = News = | ||

* July 29: The compact course is now open for registration. Please contact [mailto:d.ruprecht@leeds.ac.uk Daniel Ruprecht], [mailto:martin.schreiber@tum.de Martin Schreiber], and [mailto:neckel@in.tum.de Tobias Neckel] in one mail directly. | * July 29: The compact course is now open for registration. Please contact [mailto:d.ruprecht@leeds.ac.uk Daniel Ruprecht], [mailto:martin.schreiber@tum.de Martin Schreiber], and [mailto:neckel@in.tum.de 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 = | = 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 = | = Content = | ||

Line 24: | Line 49: | ||

* Exponential Integration | * Exponential Integration | ||

* Semi-Lagrangian methods | * 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) | ||

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

## Revision as of 12:56, 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)

- 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)

## Material

t.b.a.