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.