We consider differential problems deriving from applications in real phenomena [1, 2], where some characteristics and properties of the exact solution are a-priori known. Our aim is to develop numerical techniques that are able to preserve such features [3, 4] and that have excellent stability properties [5, 6].

In particular, we will focus on stiff differential problems, whose exact solution is positive and/or oscillates with known frequency. Numerical tests will be shown in order to confirm the efficiency, stability and accuracy of the proposed numerical methods.

References

1. Budroni, M.A., Pagano, G., Paternoster, B., D’Ambrosio, R., Conte, D., Ristori, S., Abou-Hassan, A. and Rossi, F. (2021). Synchronization scenarios induced by delayed communication in arrays of diffusively-coupled autonomous chemical oscillators. Physical Chemistry Chemical Physics. DOI: 10.1039/D1CP02221K.

2. Eigentler, L., Sherratt, J.A., (2019). Metastability as a coexistence mechanism in a model for dryland vegetation patterns. Bulletin of Mathematical Biology, 81, 2290–2322.

3. D’Ambrosio, R., Moccaldi, M., Paternoster, B., and Rossi, F. (2018). Adapted numerical modelling of the Belousov-Zhabotinsky reaction. Journal of Mathematical Chemistry, 56, 2876-2897.

4. Mickens, R.E. (2020). Nonstandard finite difference schemes: Methodology and applications. Book, 1-313.

5. Calvo, M., Montijano, J. I., Randez, L. (2021). A note on the stability of time–accurate and highly–stable explicit operators for stiff differential equations. Journal of Computational Physics. DOI: 10.1016/j.jcp.2021.110316.

6. Conte, D., D’Ambrosio, R., Pagano, G., Paternoster, B. (2020). Jacobian-dependent vs Jacobian-free discretizations for nonlinear differential problems. Computational and Applied Mathematics, 39, 171.