High order accurate time integration methods

Doctoral school course, SISSA, 2023

Notebooks of the course and instructions Course Page on SISSA website

Goal

The course aims to make the student aware of the cutting-edge algorithms used to numerically perform time integration and their properties in order to optimally choose the proper time integrator according to the considered phyisical model.

Programme

  • Theory of ODEs: examples, classification, existence and uniqueness of solutions.
  • Simple methods: explicit and implicit Euler, convergence, stability analysis, properties.
  • High order classical methods.
    • Runge–Kutta methods: construction, explicit, implicit, IMEX, error and stability analysis, properties, the Butcher tableau.
    • Multistep methods: explicit, implicit methods, error and stability analysis, convergence.
  • Iterative explicit high order methods: Deferred Correction (DeC), Arbitrary Derivative (ADER) methods, properties, stability and convergence analysis.
  • Unconditionally positivity preserving schemes: implicit Euler, high order schemes, modified Patankar schemes and their stability and convergence analysis.
  • Entropy conservative high order schemes: relaxation Runge–Kutta methods.

Prerequisites

Schedule

  • Tuesday, March 7, 2023 - 11:00 to 13:00 - Room A-132
  • Wednesday, March 8, 2023 - 11:00 to 13:00 - Room A-133
  • Thursday, March 9, 2023 - 11:00 to 13:00 - Room A-133
  • Wednesday, March 15, 2023 - 11:00 to 13:00 - Room A-133
  • Tuesday, March 21, 2023 - 10:00 to 12:00 - Room A-132 !!
  • Wednesday, March 22, 2023 - 11:00 to 13:00 - Room A-133
  • Thursday, March 23, 2023 - 11:00 to 13:00 - Room A-133
  • Tuesday, March 28, 2023 - 11:00 to 13:00 - Room A-132
  • Wednesday, March 29, 2023 - 11:00 to 13:00 - Room A-133
  • Thursday, March 30, 2023 - 11:00 to 13:00 - Room A-136