In many applications it is crucial to use numerical methods to accurately approximate physical, chemical and biological phenomena. Sometimes also some properties must be preserved, in order to have meaningful solutions. In this talk, we focus on a class of positivity-preserving time-integration methods called modified Patankar methods. It is well known that explicit schemes can be only conditionally positive (restriction on the discretization), but also high order implicit schemes are only conditionally positive. The modified Patankar Deferred Correction schemes [1] are a class of arbitrarily high order time integration methods that unconditionally preserve the positivity of the solution. They are only linearly implicit and the left-hand-side matrix is very sparse for large applications.

We will apply the scheme to some examples of ODEs and PDEs. In particular, we will see an application on shallow water equations where the spatial discretization is given by a finite volume scheme with WENO reconstruction [2]. In this case we can even obtain a computational advantage on classical (positive) explicit solvers.

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

[2] Ciallella, M., Micalizzi, L., Öffner, P., & Torlo, D. (2021). An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations. *arXiv preprint arXiv:2110.13509*.