Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

On the properties of high-order least-squares finite-volume schemes

High-order finite-volume schemes based on polynomial least-squares methods are an active research topic for the discretization of hyperbolic equations as they allow to obtain high-order spatial discretization schemes in arbitrary meshes. However, few studies have analyzed their performance in good-quality/near-to-uniform meshes, which are commonly used as a meshing strategy in zones where turbulent effects are important. In this paper, the theoretical numerical properties of commonly used least-squares (LSQ) k-exact high-order finite volume schemes are studied in one-dimensional and in several two- dimensional meshes (with some remarks regarding their properties in three-dimensional meshes). These results are compared to those obtained using fully-constrained polynomial reconstructions only compatible with structured meshes. The numerical properties of the schemes are investigated through the von Neumann analysis methodology applied to the one-dimensional and two-dimensional finite volume formulation, including temporal discretization errors. This analysis is also extended to non-uniform and unstructured two- dimensional meshes. At last, the schemes are tested with several numerical experiments using the linear advection, the Euler equations and the Navier-Stokes equations. The analysis of both studies yields similar conclusions regarding the numerical errors and stability of the different studied schemes showing that the high-order least-squares finite volume schemes yield stable and robust results across different uniform and non-uniform unstructured meshes. However, their performance is heavily degraded in simulations with low mesh resolution compared to schemes specially catered to structured meshes. On the other hand, the latter schemes lack stability and robustness in general structured meshes and its formulation cannot be straightforwardly extended to unstructured meshes. Moreover, this work shows that the use of weighted-LSQ can drastically improve the results of LSQ schemes in under-resolved simulations

Numerical Simulations of Barotropic Flows in Complex Geometries

Nella presente tesi é considerata la simulazione di flussi barotropici all'interno di geometrie complesse e le sue applicazioni a flussi cavitanti e a problemi di trasporto di sedimenti. Un approccio generale basato su vari metodi ai volumi finiti e applicabile a griglie non strutturate, é stato sviluppato e testato. L'estensione al secondo ordine spaziale é ottenuta usando metodi tipo MUSCL. L'avanzamento temporale si basa su metodi impliciti linearizzati e il secondo ordine temporale si basa sulla tecnica Defect Correction. Per quanto riguarda i flussi cavitanti, é stato utilizzato un modello a flusso omogeneo che si basa su una equazione di stato barotropica. Il bilancio di massa e di quantità di moto per flussi comprimibili sono discetizati tramite un metodo misto ai volumi finiti e agli elementi finiti. Gli elementi finiti P1 sono utilizzati per i termini viscosi mentre i volumi finiti per quelli convettivi. I flussi numerici sono calcolati utilizzando schemi in grado di calcolare soluzioni discontinue e una strategia di precondizionamento ad-hoc é stata utilizzata per risolvere i problemi di accuratezza che si riscontrano per bassi numeri di Mach. Una funzione di flusso di tipo HLL per flussi barotropici é stata proposta introdotta. In questa funzione di flusso é stato aggiunto un termine antidiffusivo che riduce i problemi di accuratezza che tipicamente si riscontrano per discontinuità di contatto e flussi viscosi quando si utilizzano schemi appartenenti a questa categoria. Per questa funzione di flusso di classe HLL due differenti linearizzazione temporali sono state considerate: nella prima la matrice di upwind della funzione di flusso é considerata constante, mentre nella seconda la sua variazione temporale viene tenuta in considerazione. Gli ingredienti numerici proposti sono stati quindi testati simulando varie tipologie di flussi, in particolare lo strato limite di Blasius, un problema di Riemann, il flusso quasi-1D in un ugello e il flusso di acqua intorno ad un profilo, sia in condizioni cavitanti che non cavitanti. Inoltre l'introduzione degli effetti della turbolenza tramite il modello RANS k-epsilon é stata testata simulando il flusso ad alto numero di Reynolds su una lastra piana e, per finire, é stata affrontata la simulazione numerica di un induttore reale tridimensionale, sia in condizioni non cavitanti e cavitanti. Oltre a quanto detto sono stati considerati anche problemi di trasporto di sedimenti. Il modello fisico di questo problema é basato sulle equazioni Shallow-Water a cui si aggiunge l'equazione di Exner per descrivere l'evoluzione temporale del profilo del fondale. In particolare é il flusso di sedimenti é stato descritto utilizzando il modello di Grass. Il sistema completo di equazioni é stato discretizzato utilizzando due metodi ai volumi finiti, lo schema di tipo predittore-correttore SRNH e uno schema di Roe modificato per sistemi di equazioni in forma non conservativa. Partendo dalle versioni esplicite di questi schemi, sono stati sviluppati i corrispondenti metodi impliciti e, in particolare lo Jacobiano della funzione di flusso é stato calcolato utilizzando strumenti di differenziazione automatica. Questo approccio permette di non dover calcolare manualmente le complesse espressioni delle derivate della funzione di flusso. Questi metodi sono poi stati comparati in termini di accuratezza e costi computazionali utilizzando specifici problemi monodimensionali e bidimensionali caratterizzati da scale temporali diverse per l'evoluzione del fondale e del flusso d'acqu

Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities

In simulating compressible flows with contact discontinuities or material interfaces, numerical pressure and velocity oscillations can be induced by point-wise flux vector splitting (FVS) or component-wise nonlinear difference discretization of convection terms. The current analysis showed that the oscillations are due to the incompatibility of the point-wise splitting of eigenvalues in FVS and the inconsistency of component-wise nonlinear difference discretization among equations of mass, momentum, energy, and even fluid composition for multi-material flows. Two practical principles are proposed to prevent these oscillations: (i) convective fluxes must be split by a global FVS, such as the global Lax-Friedrichs FVS, and (ii) consistent discretization between different equations must be guaranteed. The latter, however, is not compatible with component-wise nonlinear difference discretization. Therefore, a consistent discretization method that uses only one set of common weights is proposed for nonlinear weighted essentially non-oscillatory (WENO) schemes. One possible procedure to determine the common weights is presented that provided good results. The analysis and methods stated above are appropriate for both single- (e.g., contact discontinuity) and multi-material (e.g., material interface) discontinuities. For the latter, however, the additional fluid composition equation should be split and discretized consistently for compatibility with the other equations. Numerical tests including several contact discontinuities and multi-material flows confirmed the effectiveness, robustness, and low computation cost of the proposed method. (C) 2015 Elsevier Inc. All rights reserved

Object-oriented hyperbolic solver on 2D-unstructured meshes applied to the shallow water equations

Fluid dynamics, like other physical sciences, is divided into theoretical and experimental branches. However, computational fluid dynamics (CFD) is third branch of Fluid dynamics, which has aspects of both the previous two branches. CFD is a supplement rather than a replacement to the experiment or theory. It turns a computer into a virtual laboratory, providing insight, foresight, return on investment and cost savings1. This work is a step toward an approach that realise a new and effective way of developing these CFD models

Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas

Ideal GLM-MHD - a new mathematical model for simulating astrophysical plasmas

Magnetic fields are ubiquitous in space. As there is strong evidence that magnetic fields play an important role in a variety of astrophysical processes, they should not be neglected recklessly. However, analytic models in astrophysical either do often not take magnetic fields into account or can do this after limiting simplifications reducing their overall predictive power. Therefore, computational astrophysics has evolved as a modern field of research using sophisticated computer simulations to gain insight into physical processes. The ideal MHD equations, which are the most often used basis for simulating magnetized plasmas, have two critical drawbacks: Firstly, they do not limit the growth of numerically caused magnetic monopoles, and, secondly, most numerical schemes built from the ideal MHD equations are not conformable with thermodynamics. In my work, at the interplay of math and physics, I developed and presented the first thermodynamically consistent model with effective inbuilt divergence cleaning. My new Galilean-invariant model is suitable for simulating magnetized plasmas under extreme conditions as those typically encountered in astrophysical scenarios. The new model is called the "ideal GLM-MHD" equations and supports nine wave solutions. The accuracy and robustness of my numerical implementation are demonstrated with a number of tests, including comparisons to other schemes available within in the multi-physics, multi-scale adaptive mesh refinement (AMR) simulation code FLASH. A possible astrophysical application scenario is discussed in detail