High order accurate time integration methods

Doctoral school course, Ecole doctorale n°39, Université de Bordeaux, 2021

Notebooks of the course and instructions Course Page on ADUM website


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.


  • 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.



  • Session 1: 14/04/2021 9:00-12:00
  • Session 2: 14/04/2021 13:30-16:30
  • Session 3: 15/04/2021 9:00-12:00
  • Session 4: 15/04/2021 13:30-16:30