Continuous Galerkin high order well-balanced discrete kinetic model for shallow water equations
Talk, Numhyp 2021, Trento, Italy
Kinetic models describe many physical phenomena, inter alia Boltzmann equations, but can also be used to approximate with an artificial relaxation procedure other macroscopic models. We consider the kinetic model proposed by Aregba-Driollet and Natalini \cite{natalini}, and we modify it in order to approximate shallow water (SW) equations. The difference with the original model stands in the presence of the source term in the SW equations due to the effect of the bathymetry. Thus, the kinetic model \cite{natalini} must be extended in order to include this term and to maintain the asymptotic convergence to the macroscopic limit of the SW problem.
On modified Patankar schemes and oscillations: towards new stability definitions
Talk, Icosahom 2020, Vienna, Austria
Modified Patankar (MP) schemes are linearly implicit ODE solvers for production destruction problems that guarantee unconditionally the positivity of the solutions and the conservation of the total quantities.
ADER and DeC: Arbitrarily High Order Explicit Methods for hyperblic PDEs and ODEs
Talk, Oberwolfach, workshop on hyperbolic balance laws, Oberwolfach, Germany
The hyperbolic PDE community has used, in the last years, two powerful techniques that provide arbitrarily high order explicit schemes: ADER (arbitrary derivative) schemes and Deferred Correction (DeC) schemes.
High Order Well-Balanced Discrete Kinetic Model for Shallow Water Equations
Talk, WCCM - Eccomas 2020, Paris, France
In this work, we study a kinetic model that contains stiff relaxation terms in the source. This model can be applied to any non linear hyperbolic problem without source term, that we will call macroscopic problem, to obtain a larger system of equations with linear fluxes and non–linear source terms, the microscopic problem, that converges asymptotically to the original hyperbolic system.
Arbitrary high-order, conservative and positive preserving Patankar-type deferred correction schemes
Talk, Icosahom 2020 MS Online, Online
Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.
Hyperbolic problems: high order methods and model order reduction
Talk, University of Zurich, Zurich, Switzerland
PhD Defense at UZH Zurich. The talk summarize the main topics of the thesis: high order methods for ODEs, hyperbolic PDEs and model order reduction techniques.
Model Reduction for Advection Dominated Hyperbolic Problems in an ALE Framework: Offline and Online Phases
Talk, Analysis Junior Seminars 2019-2020, Trieste, Italy
Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems.
ADER and DeC: Arbitrarily High Order Explicit Time Integration Methods
Talk, SAMinar (ETH+UZH), Zurich, Switzerland
When we think about high order time integration, Runge–Kutta (RK) is the most known class of schemes that comes to mind, for historical reasons. Nevertheless, many other techniques have been developed during these years to improve these techniques and to get a generalized form of them.