Projective multiscale time-integration for electrostatic particle-in-cell methods

The simulation of problems in kinetic plasma physics is often challenging due to strongly coupled phenomena across multiple scales. In this work, we propose a wavelet-based coarse-grained numerical scheme, based on the framework of Equation-Free Projective Integration, for a kinetic plasma system modeled by the Vlasov–Poisson equations. A kinetic particle-in-cell (PIC) code is used to simulate the meso scale dynamics for short time intervals. This allows the extrapolation over long time-steps of the behavior of a coarse wavelet-based discretization of the system. To validate the approach and the underlying concepts, we perform two 1D1V numerical experiments: nonlinear propagation and steepening of an ion wave, and the expansion of a plasma slab in vacuum. The direct comparisons to resolved PIC simulations show good agreement. We show that the speedup of the projective integration scheme over the full particle scheme scales linearly with the system size, demonstrating efficiency while taking into account fully kinetic, non-Maxwellian effects. This suggests that the approach is potentially interesting for kinetic plasma problems with a large separation of scales.

An adaptive Rosenbrock method with applications in Electrophysiology

Giacomo Rosilho De Souza

The aim of this project is to implement the Rosenbrock method ROS3P in the C++ Finite Element library LifeV for the solution of systems of ordinary differential equations arising in electrophysiology. In the domain of electrophysiology LifeV implements cardiac cell models which can require the resolution of stiff reaction-diffusion equations. To solve this system the use of implicit methods is required. We focus in particular on one step methods. Implicit methods need the solution of a non linear system at each time step. The Rosenbrock methods derive from the linearly implicit Runge Kutta methods and just requires the solution of s linear systems per time step, where s is the number of stages. Most of the implicit Runge Kutta methods suffer from order reduction when applied to parabolic problems. For this reason we have chosen the third order Rosenbrock method ROS3P, which satisfies additional order conditions to keep the same convergence order even when applied to non linear parabolic problems. In the first part of this work we derive the Rosenbrock methods from Runge Kutta methods, explaining the time adaptivity strategy and finally giving the algorithm of our implementation in LifeV. After we present the cardiac reaction-diffusion models, deriving the mono-domain model from the bi-domain model. In the following we explain the strategy that we adopt to solve these models and how we discretize them. Finally we show some numerical experiments obtained with our approach and compare our results with the ones given by other methods.

2013

Coupled Mathematical Models for Heart Integration: A Stability Study

Andrea Di Blasio

In this thesis we consider a fully coupled model which aims at reproducing some qualitative features of the electro-mechanical activity of the heart. The models used to describe both the electrical and mechanical activities are relatively simple. However, coupling them together can give rise to numerical instabilities or incorrect predictions. After having introduced each of the sub-models of the fully coupled system we perform some numerical experiments to draw some insights on the numerical approximation of this problem. Firstly we focus on the numerical approximation of the Aliev-Panfilov model, which controls the electrical activation of the muscle. We verify that different approaches can be followed to solve such a problem by the finite element method reducing the computational effort. However each approach can lead to inaccurate predictions of the front velocity. Then we suggest also two numerical schemes for time integration particularly suited for PDEs such as the Aliev-Panfilov model: the operator splitting method and the Runge-Kutta-Chebyshev method. When considering the fully coupled problem, we examine two ways of reducing the computational cost: treating some of the coupling terms explicitly or solving the linearised system iteratively. We verify that with the first choice we can experience numerical instabilities depending on the numerical scheme used for time integration. On the other hand, when solving the linearised system iteratively, key points to solve the problem efficiently are the choice of an adaptive stopping criterion and a good preconditioner. From the numerical experiments performed we conclude that the coupling between the active stress and the mechanics is very influential on the stability of the system and on the convergence of the residuals.

2014

Sur quelques équations aux dérivées partielles en lien avec le lemme de Poincaré

David Valentin Strütt

In this thesis, we study two distinct problems. The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system. The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

EPFL2018