IMplicit-EXplicit Formulations for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models

This work presents IMplicit-EXplicit (IMEX) formulations for discontinuous Galerkin (DG) discretizations of the compressible Euler equations governing non-hydrostatic atmospheric flows. In particular, we show two different IMEX formulations that not only treat the stiffness due to the governing dynamics but also the domain discretization. We present these formulations for two different equation sets typically employed in atmospheric modeling. For both equation sets, efficient Schur complements are derived and the challenges and remedies for deriving them are discussed. The performance of these IMEX formulations of different orders are investigated on both 2D (box) and 3D (sphere) test problems and shown to achieve their theoretical rates of convergence and their efficiency with respect to both mesoscale and global applications are presented

"Mariage des Maillages": A new numerical approach for 3D relativistic core collapse simulations

We present a new 3D general relativistic hydrodynamics code for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. It uses spectral methods for solving the metric equations, assuming the conformal flatness approximation for the three-metric. The matter equations are solved by high-resolution shock-capturing schemes. We demonstrate that the combination of a finite difference grid and a spectral grid can be successfully accomplished. This "Mariage des Maillages" (French for grid wedding) approach results in high accuracy of the metric solver and allows for fully 3D applications using computationally affordable resources, and ensures long term numerical stability of the evolution. We compare our new approach to two other, finite difference based, methods to solve the metric equations. A variety of tests in 2D and 3D is presented, involving highly perturbed neutron star spacetimes and (axisymmetric) stellar core collapse, demonstrating the ability to handle spacetimes with and without symmetries in strong gravity. These tests are also employed to assess gravitational waveform extraction, which is based on the quadrupole formula.Comment: 29 pages, 16 figures; added more information about convergence tests and grid setu

Local preconditioning for parallel iterative solvers

This thesis aims at improving the convergence of iterative solvers, used for algebraic systems coming from the discretization of partial differential equations (PDE), in the context of large scale simulations and high performance computing (HPC). The methodology followed consists in adapting some existing preconditiong techniques to the physics and numerics of convection-dominated transport and boundary layer problems in flows. For convection-dominated flows, a physics-based permutation algorithm is presented, which consists in renumbering the mesh in the direction of convection. This renumbering is then used together with a Gauss-Seidel preconditioner to propagate the result of the matrix-vector products along the convection. The robutsness and effectiveness of this preconditioner is proved in several test cases solving the heat equation as well as the Navier-Stokes equations in both sequential and in parallel using the Message Passing Interface library MPI. Additionally, the composition of preconditioners is proposed to solve cases where different local physical behaviors co-exist in the same flow. In particular, we focus on such problems where of a highly convective flow encounters an obstacle. Such problems involve a zone with high convection far from the obstacle and the development of a boundary layer in the vicinity of the obstacle. In numerical terms, these local behaviors translate into specific matrix structures that we will take advantage of to adapt the preconditioner locally. On the one hand, the linelet preconditioner is a well-known efficient preconditioner for boundary layers where the mesh is highly anisotropic, in particular to solve the Poisson equation. On the other hand, the streamline linelet that we propose in this thesis (Gauss-Seidel together with a mesh renumbering in the convection direction) is well adapted for locally hyperbolic flows. Both preconditioners will be composed (combined) in different ways to investigate their robustness in terms of convergence as well as their costs to solve the proposed transport problems. We will study as well their performances in terms of parallelization.Esta tesis tiene como objetivo mejorar la convergencia de los métodos iterativos utilizados para resolver sistemas de ecuaciones algebraicas provenientes de la discretización de ecuaciones diferenciales en derivadas parciales (EDP), en el contexto de las simulaciones a gran escala y computación de altas prestaciones (HPC). La metodología seguida consiste en adaptar algunas técnicas de precondicionamiento existentes, a la física y la numérica en flujos que presentan una alta convección y flujos que presentan una capa límite. Para los flujos dominados por convección, se presenta un algoritmo de permutación basado en la física, que consiste en la renumeración de la malla en la dirección de la convección. Esta renumeración se usa luego junto con el precondicionador Gauss-Seidel para propagar el resultado de los productos matriz-vector a lo largo de la convección. La robustez y eficiencia de este precondicionador se demuestra en varios ejemplos en los que se resuelve la ecuación de calor y las ecuaciones de Navier-Stokes tanto en secuencial como en paralelo utilizando la librería interfaz de paso de mensajes (MPI). Además, se propone la composición de precondicionadores para resolver casos donde diferentes comportamientos físicos locales coexisten en el mismo flujo. En particular, nos enfocamos en los casos donde un flujo altamente convectivo se encuentra un obstáculo. En este tipo de problemas nos encontramos dos zonas: una con alta convección lejos del obstáculo y otra donde se desarrolla una capa límite en los alrededores del obstáculo. En términos numéricos, estos comportamientos locales se traducen en estructuras matriciales específicas que aprovecharemos para adaptar localmente el precondicionador. Por un lado, sabemos que el linelet es un precondicionador eficiente para resolver problemas de capa límite donde la malla es altamente anisótropa. En particular resulta eficiente para resolver la ecuación de Poisson. Por otro lado, sabemos que el linelet aerodinámico, el precondicionador que proponemos en esta tesis (precondicionador Gauss-Seidel junto con una renumeración de malla en la dirección de la convección) está bien adaptado para flujos localmente hiperbólicos. Con todo esto, proponemos también una composición de los dos precondicionadores (combinación de ambos) de distintas formas para investigar su robustez en términos de convergencia, así como sus costes para resolver los problemas de transporte propuestos. Estudiaremos también el rendimiento en cuanto a la paralelización se refiere.Postprint (published version

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte