Ordinary differential equations (ODEs) describe many physical, biological and chemical phenomena. It is, thus, important to find approximations of ODEs which are highly accurate and, in order to obtain it within reasonable computational times, high order accurate time integration methods are very often chosen to proceed in time. In this course, we will revise ODEs and the theoretical results that guarantee their existence and uniqueness [1, 2]. Then, we will overview classical solver, starting from low order ones, going to high order, with a particular attention on Runge–Kutta and Multistep methods [1, 3]. We will see different variants of these schemes and their properties (order of accuracy, stability, structure preserving) with a particular focus on the distinction between implicit and explicit methods and their combination in the IMEX methods [2, 7]. We will further introduce the Deferred Correction [4, 5, 6] and the Arbitrary Derivative [8, 10] methods, which are iterative arbitrarily high order explicit, implicit and IMEX time integration methods. Finally, we will see how these algorithms can be modified in order to guarantee the conservation of some physical properties, such as positivity of some quantities [9] or conservation of the total entropy or energy [11].

The course will be in a hybrid form with theoretical sessions alternated with hands-on sessions. Minimal knowledge of programming is required.

Course materials: https://github.com/accdavlo/HighOrderODESolvers

Program

1. Review of theory of ODEs: examples, classification, existence and uniqueness of solutions.

2. Simple methods: explicit and implicit Euler, convergence, stability analysis, properties.

3. High order classical methods:

• Runge–Kutta methods: construction, explicit, implicit, IMEX, collocation methods, error and stability analysis, properties, the Butcher tableau.

• Multistep methods: explicit, implicit methods, error and stability analysis, convergence.

4. Iterative explicit high order methods: Deferred Correction (DeC), Arbitrary Derivative (ADER) methods, properties, stability and convergence analysis.

5. Unconditionally positivity preserving schemes: implicit Euler, high order schemes, modified Patankar schemes and their stability and convergence analysis.

6. Entropy conservative high order schemes: relaxation Runge–Kutta methods.

References

[1] Hairer, E. and Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg, 1993.

[2] Wanner, G. and Hairer, E., Solving Ordinary Differential Equations II. Stiff and Algebraic Problems. Vol. 375. New York: Springer Berlin Heidelberg, 1996.

[3] Butcher, J. C., Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Ltd, 2008.

[4] R. Abgrall. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. Journal of Scientific Computing, 73(2):461–494, 2017.

[5] A. Dutt, L. Greengard and V. Rokhlin. Spectral Deferred Correction Methods for Ordinary Differential Equations. BIT Numerical Mathematics, 40(2):241–266, 2000.

[6] M. Minion. Semi-implicit spectral deferred correction methods for ordinary differential equations. Communication in Mathematical Physics 1(3):471–500, 2003.

[7] S. Boscarino, L. Pareschi and G. Russo, Implicit-Explicit Runge–Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit. SIAM Journal on Scientific Computing, 35(1):A22–A51, 2013.

[8] Dumbser, M. and Balsara, D. S. and Toro, E. F. and Munz, C.-D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. Journal of Computational Physics, 227(18):8209–8253, 2008.

[9] Öffner, P. and Torlo, D., Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes. Applied Numerical Mathematics, 153:15-34, 2020.

[10] Han Veiga, M. and Öffner, P. and Torlo, D., DeC and ADER: Similarities, Differences and a Unified Framework. Journal of Scientific Computing, 87(1):1–35, 2021.

[11] Ranocha, H. and Sayyari, M. and Dalcin, L. and Parsani, M. and Ketcheson, D. I., Relaxation Runge–Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier–Stokes Equations. SIAM J. Sci. Comput., 42(2), A612–A638, 2020.