Discontinuous Galerkin Spectral Element Methods for Astrophysical Flows in Multi-physics Applications

In engineering applications, discontinuous Galerkin methods (DG) have been proven to be a powerful and flexible class of high order methods for problems in computational fluid dynamics. However, the potential benefits of DG for applications in astrophysical contexts is still relatively unexplored in its entirety. To this day, a decent number of studies surveying DG for astrophysical flows have been conducted. But the adoption of DG by the astrophysics community is just beginning to gain traction and integration of DG into established, multi-physics simulation frameworks for comprehensive astrophysical modeling is still lacking. It is our firm believe, that the full potential of novel approaches for numerically solving the fluid equations only shows under the pressure of real-world simulations with all aspects of multi-physics, challenging flow configurations, resolution and runtime constraints, and efficiency metrics on high-performance systems involved. Thus, we see the pressing need to propel DG from the well-trodden path of cataloguing test results under "optimal" laboratory conditions towards the harsh and unforgiving environment of large-scale astrophysics simulations. Consequently, the core of this work is the development and deployment of a robust DG scheme solving the ideal magneto-hydrodynamics equations with multiple species on three-dimensional Cartesian grids with adaptive mesh refinement. We chose to implement DG within the venerable simulation framework FLASH, with a specific focus on multi-physics problems in astrophysics. This entails modifications of the vanilla DG scheme to make it fit seamlessly within FLASH in such a way that all other physics modules can be naturally coupled without additional implementation overhead. A key ingredient is that our DG scheme uses mean value data organized into blocks - the central data structure in FLASH. Having the opportunity to work on mean values, allows us to rely on a rock-solid, monotone Finite Volume (FV) scheme as "backup" whenever the high order DG method fails in cases when the flow gets too harsh. Finding ways to combine the two schemes in a fail-safe manner without loosing primary conservation while still maintaining high order accuracy for smooth, well-resolved flows involves a series of careful considerations, which we document in this thesis. The result of our work is a novel shock capturing scheme - a hybrid between FV and DG - with smooth transitions between low and high order fluxes according to solution smoothness estimators. We present extensive validations and test cases, specifically its interaction with multi-physics modules in FLASH such as (self-)gravity and radiative transfer. We also investigate the benefits and pitfalls of integrating end-to-end entropy stability into our numerical scheme, with special focus on highly compressible turbulent flows and shocks. Our implementation of DG in FLASH allows us to conduct preliminary yet comprehensive astrophysics simulations proving that our new solver is ready for assessments and investigations by the astrophysics community

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

On the Advective Component of Active Flux Schemes for Nonlinear Hyperbolic Conservation Laws

A new class of numerical methods called Active Flux (AF) is investigated for nonlinear hyperbolic conservation laws. The AF method is designed specifically to address the aspect that most modern compressible flow methods fail to do; the multidimensionality aspect. It addresses the shortcoming by employing a two stage update process. In the first stage, a nonconservative form of the system is introduced to provide the flexibility to pursue distinct numerical approaches for flow processes with differing physics. Because each process is treated separately, the numerical method can be appropriately formed to reflect each type of physics and to provide the maximal stability. The method is completed with the conservation update to produce a third-order accurate scheme. The AF advection scheme is founded on the characteristic tracing method, a semi-Lagrangian method, which has long been used for developing numerical methods for hyperbolic problems. The first known AF method for advection, Scheme V by van Leer, is revisited as a part of the development of the scheme. Details of Scheme V are examined closely, and new improvements are made for the multidimensional nonlinear advection scheme. A detailed study of the nonlinear system of equations is made possible by the pressureless Euler system, which is the advective component of the Euler system. It serves as a stepping stone for the Euler system, and all necessary details of the nonlinear system are explored. Lastly, an extension to the Euler system is presented where a novel nonlinear operator splitting method is introduced to correctly blend the contributions of the nonlinear advection and acoustic processes. The AF method, as a result, produces a maximally stable, third-order accurate method for the multidimensional Euler system. Some guiding principles of limiting are presented. Because two types of flow feature are separately treated, the limiting process must also be kept separate. Advective problems obeying natural bounding principles are treated differently from acoustic problems with no explicit bounding principles. Distinct limiting approaches are explored along with discussions.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138695/1/jmaeng_1.pd

Dinâmica espacial e temporal do metabolismo aquático em sistemas subtropicais

A tese investigou o uso de uma ferramenta computacional para avaliar a dinâmica espaço-temporal do metabolismo em ecossistemas aquáticos onde a hidrodinâmica possui papel de destaque na estruturação das comunidades planctônicas. A modelagem matemática foi realizada utilizando o modelo IPH-ECO, uma ferramenta computacional complexa capaz de integrar processos físicos, químicos e biológicos em três dimensões. O trabalho foi dividido em quatro capítulos principais, tendo como base os processos físicos e biológicos que influenciam as estimativas de metabolismo (Capítulos #02 e #03) e melhorias nos métodos numéricos utilizados no modelo IPH-ECO (Capítulos #04 e #05). Os primeiros dois capítulos apresentam o desenvolvimento e aplicação de um algoritmo computacional capaz de quantificar as estimativas de metabolismo aquático baseado (em termos de Produção Primária Bruta - GPP, Respiração do ecossistema - R e Produção Líquida do Ecossistema - NEP = GPP - R) em processos biológicos individuais que influenciam o balanço de oxigênio dissolvido em ecossistemas aquáticos (e.g., respiração de zooplâncton, produção primária de macrófitas aquáticas). A implementação deste algoritmo no modelo IPH-ECO permitiu quantificar as estimativas de metabolismo aquático na Lagoa Mangueira, sul do Brasil, avaliando a importância relativa de diferentes processos individuais e o efeito da hidrodinâmica sobre os processos que compõem o metabolismo da lagoa. O metabolismo aquático da Lagoa Mangueira apresentou um gradiente espacial com maiores valores na região Litorânea e menores na região Pelágica. Além da heterogeneidade espacial, foi possível observar uma heterogeneidade temporal, com valores de produção primária mais elevados durante o verão e primavera e menores durante o inverno e outono. Esta heterogeneidade espacial e temporal acarreta em alterações no estado trófico (autotrófico - NEP positivo ou heterotrófico - NEP negativo) da lagoa, dependendo do local sendo avaliado (zona litorânea ou zona pelágica) e da época do ano. A simulação de diferentes cenários de vento (cinco no total) demonstraram que os padrões de circulação da água podem alterar a dinâmica das estimativas de metabolismo na Lagoa Mangueira, alterando a forma como o sistema é classificado (autotrofia vs. heterotrofia) e influenciando os diferentes processos biológicos que compõem estas estimativas. Os Capítulos #04 e #05 apresentam o desenvolvimento de um novo esquema numérico visando auxiliar na simulação de problemas de qualidade de água. O novo esquema é baseado no método dos Volumes Finitos e permite a integração numérica de equações de transporte utilizando um passo de tempo localizado, calculado a partir da condição de estabilidade de Courant-Friedrich-Lewy (condição CFL). A nova solução numérica é diretamente acoplada a um modelo hidrodinâmico tridimensional em grades triangulares não-estruturadas (e.g, modelo UnTRIM), que utiliza uma solução numérica semi-implícita (Crank-Nicholson) baseada em diferenças finitas e volumes finitos. Diferentes testes clássicos e idealizados são simulados e é realizada uma comparação entre o método com esquema numérico localizado (LTS - Local Time Stepping) e o método tradicional (GTS - Global Time Stepping). Ambos os métodos se mostraram conservativos considerando uma, duas e três dimensões, e ainda foi respeitada uma condição de estabilidade baseada nos valores máximos e mínimos sendo transportados (i.e., não são criados novos valores máximos nem mínimos). Os métodos também foram avaliados de forma acoplada com escoamentos a superfície livre, levando em conta substâncias conservativas e não-conservativas (e.g., balanço de temperatura na água), assim como situações onde a hidrodinâmica é controlada por vento (forte mistura vertical e horizontal) e onde a hidrodinâmica é controlada por um gradiente de pressão (e.g., maré). Além disso, situações onde a secagem e inundação de células computacionais ocorrem foram testadas e os métodos se mostraram estáveis e conservativos. O esquema numérico LTS se mostrou mais rápido do ponto de vista computacional, exigindo menos tempo de simulação em praticamente todos os testes realizados. Além disso, o esquema mostrou resultados similares ao obtidos utilizando o esquema GTS tradicional. Os testes mostraram que a eficiência do esquema LTS é maior quando ocorre a combinação de altas velocidades e pequenos elementos (alta restrição dada pela condição CFL), como a simulação da interface entre rios e lagos, entradas de água rápida (e.g., tromba d’água, Dam-Break) e estuários (efeito de maré).This thesis investigated the use of a mathematical model to evaluate the space-time dynamics of aquatic metabolism in ecossystems where hydrodynamics plays a key role in structuring the planktonic community. The mathematical model used was the IPH-ECO model, a complex tridimensional model capable of integrating physical, chemical, and biological processes in aquatic environments. The thesis was divided into four main chapters with focus in studying the biological processes and aquatic metabolism estimates (Chapters #2 and #3), and also the improvement of numerical methods used in the IPH-ECO model (Chapters #4 and #5). The first two chapters show the development and application of a computational algorithm capable of quantifying the aquatic metabolism estimates (in terms of Gross Primary Production - GPP, Ecosystem Respiration - R, and Net Ecosystem Production - NEP = GPP-R) based on individual biological processes affecting the dissolved oxygen budget in aquatic ecosystems (e.g., zooplankton respiration, aquatic macrophyte primary production). The numerical algorithm implemented on the IPH-ECO model allowed the quantification of aquatic metabolism estimates (GPP, R, and NEP) at Lake Mangueira, South of Brazil, and the evaluation of the relative importance of different individual processes and how the lake hydrodynamic can change the complex dynamics of the biological processes comprising the lake metabolism estimates. Lake Mangueira’s metabolism estimations showed a well-marked spatial gradient with higher values observed in the littoral zone and lower values observed in the pelagic zone. Besides the spatial heterogeneity, it was also possible to notice a strong seasonal heterogeneity, with increased values of GPP and R during summer and spring and lower values during winter and autumn. This space-time heterogeneity leads to a switching in the trophic state of the lake (autotrophic - Positive NEP or heteotrophic - Negative NEP), depending on the site being evaluated (littoral or pelagic) and also the time of the year being assessed. The simulation of different wind scenarios (a total of five) showed that the water circulation patterns can change the metabolism estimates dynamics in Lake Mangueira, changing the system trophic status (net autotrophy v.s. net heterotrophy) e also affecting the dynamics of individual biological processes composing the metabolism estimates. Chapters #4 and #5 show the development of a new numerical scheme capable of accelerate water quality simulations. The new numerical method is based on a Finite Volume framework and allows for numerical integration of scalar transport equations using a local time step, chosen based on the Courant-Friedrich-Lewy stability criteria (the CFL condition). This new solution is directly linked to a tridimensional hydrodynamic model on triangular unstructured mesh (e.g., UnTRIm model), using a semi-implicit solution based on finite differences and finite volume. Different classical and idealized test-cases were simulated and the results from using the new Local Time Stepping (LTS) numerical method is compared against the usage of a traditional Global Time Stepping method (GTS). Both implemented methods showed precise mass conservation in one, two, and three dimensions, moreover, a discrete max-min property was observed in all simulations (i.e., no new maximum nor minimum was created). The methods were also tested when the coupling with hydrodynamic models of free-surface flows is simulated, accounting both conservative (e.g., Salt) and non-conservative substances (e.g., water temperature). The idealized coupled test-cases accounted for situations where hydrodynamics is controled by the wind (strong vertical and horizontal mixing), and situations where hydrodynamics is driven by pressure gradients (e.g., tidal currents). Furthermore, situations where an intense wet- and dry-ing of computational cells is observed was tested and the methods showed stability and precise mass conservation. In a general manner, the new LTS scheme was faster from a computational point-of-view, requiring less simulation time in praticaly all tests. In addition, the new scheme presented concentration fields similar to the ones computed by a traditional GLS algorithm. Our findings suggested that the efficiency of the LTS algorithm is increased when a combination of high velocities and small polygons is observed (elevated CFL stability restriction), such as the simulation of the interface between rivers and lakes, fast water inflow (e.g., Dam-Break), and estuaries (Tidal effect)

High-Order Finite-Volume Schemes for Magnetohydrodynamics

New high-order finite-volume numerical schemes for the magnetohydrodynamics equations are proposed in two and three dimensions. Two different sets of magnetohydrodynamics equations are considered. The first set is the ideal magnetohydrodynamics system, which assumes that the fluid can be treated as a perfect conductor. The second set is resistive MHD, which involves non-zero resistivity. A high-order central essentially nonoscillatory (CENO) approach is employed, which combines unlimited k-exact polynomial reconstruction with a monotonicity preserving scheme. The CENO schemes, which were originally developed for compressible fluid flow, are applied to the MHD equations, along with two possible control mechanisms for divergence error of the magnetic field. The hyperbolic fluxes are calculated by solving a Riemann problem at each cell interface, and elliptic fluxes are computed through k-exact gradient interpolation where point-wise values of the gradients are required. Smooth test problems and test cases with discontinuities (weak or strong) are considered, and convergence studies are presented for both the ideal and resistive MHD systems. Several potential space physics applications are explored. For these simulations, cubed-sphere grids are used to model the interaction of the solar wind with planetary bodies or their satellites. The basic cubed-sphere grid discretizes a simulation domain between two concentric spheres using six root blocks (corresponding to the six faces of a cube). Conditions describing the atmosphere of the inner body can be applied at the boundary of the inner sphere. For some problems we also need to solve equations within the inner sphere, for which we develop a seven-block cubed-sphere grid where the empty space inside the interior sphere is discretized as a seventh root block. We consider lunar flow problems for which we employ the seven-block cubed-sphere mesh. Ideal MHD is solved between the inner and outer spheres of the grid, and the magnetic diffusion equations are solved within the inner sphere, which represents the lunar interior. Two cases are considered: one is without intrinsic magnetic field, where only a wake is expected without any bow shock forming ahead of the Moon, and the second is with a small dipole moment to model a lunar crustal magnetic anomaly, in which case a small-scale magnetosphere is expected ahead of the region with the magnetic anomaly