AMR vs High Order Schemes Wavelets as a Guide

The final goal behind any numerical method is give the smallest wall-clock time for a given final time error or, conversely, the smallest run-time error for a given wall clock time, etc. Here a comparison will be given between adaptive mesh refinement schemes and non-adaptive schemes of higher order. It will be shown that in three dimension calculations that in order for AMR schemes to be competitive that the finest scale must be restricted to an extremely, and unrealistic, small percentage of the computational domain

AMR, stability and higher accuracy

Efforts to achieve better accuracy in numerical relativity have so far focused either on implementing second order accurate adaptive mesh refinement or on defining higher order accurate differences and update schemes. Here, we argue for the combination, that is a higher order accurate adaptive scheme. This combines the power that adaptive gridding techniques provide to resolve fine scales (in addition to a more efficient use of resources) together with the higher accuracy furnished by higher order schemes when the solution is adequately resolved. To define a convenient higher order adaptive mesh refinement scheme, we discuss a few different modifications of the standard, second order accurate approach of Berger and Oliger. Applying each of these methods to a simple model problem, we find these options have unstable modes. However, a novel approach to dealing with the grid boundaries introduced by the adaptivity appears stable and quite promising for the use of high order operators within an adaptive framework

Propagation of Periodic Waves Using Wave Confinement

This thesis studies the behavior of the Eulerian scheme, with Wave Confinement (WC), when propagating periodic waves. WC is a recently developed method that was derived from the scheme vorticity confinement used in fluid mechanics, and it efficiently solves the linear wave equation. This new method is applicable for numerous simulations such as radio wave propagation, target detection, cell phone and satellite communications. The WC scheme adds a nonlinear term to the discrete wave equation that adds stability with negative and positive diffusion, conserves integral quantities such as total amplitude and wave speed, and it allows wave propagation over long distances with minimal numerical diffusion, which contrasts to other numerical methods where wave propagation is affected by numerical dissipation. Previous studies have shown that WC propagates short pulses/surfaces as thin nonlinear solitary waves. In this thesis, a one-dimensional (1D) periodic wave is propagated by WC using the advection and wave equations. For the advection equation, the parameters and the initial condition (IC) used in WC are analyzed to establish for which conditions the method can be implemented. When the IC is a positive periodic wave, the converged solution consists of a series of hyperbolic secants where the number of cycles of the IC represents the number of hyperbolic secants. Waves with varying signs are analyzed by changing the wave confinement term. For this case, the converged solution is a series of positive and negative hyperbolic secants where each hyperbolic secant is represented by half cycle of the IC. For the wave equation, parameters and different IC\u27s are studied to determine when WC is feasible. For positive periodic waves, the converged solution retains its sinusoidal shape and does not converge to a series of hyperbolic secants. The waves with varying signs, however, converge to a series of hyperbolic secants as seen for the advection equation. WC is stable for various periodic waves for both advection and wave equations, which shows WC is useful for numerically propagating periodic waveforms. Convergence depends on the wave number of the IC and on the parameters (convection speed, positive diffusion, negative diffusion) used in WC

An Application of Gaussian Process Modeling for High-order Accurate Adaptive Mesh Refinement Prolongation

We present a new polynomial-free prolongation scheme for Adaptive Mesh Refinement (AMR) simulations of compressible and incompressible computational fluid dynamics. The new method is constructed using a multi-dimensional kernel-based Gaussian Process (GP) prolongation model. The formulation for this scheme was inspired by the GP methods introduced by A. Reyes et al. (A New Class of High-Order Methods for Fluid Dynamics Simulation using Gaussian Process Modeling, Journal of Scientific Computing, 76 (2017), 443-480; A variable high-order shock-capturing finite difference method with GP-WENO, Journal of Computational Physics, 381 (2019), 189-217). In this paper, we extend the previous GP interpolations and reconstructions to a new GP-based AMR prolongation method that delivers a high-order accurate prolongation of data from coarse to fine grids on AMR grid hierarchies. In compressible flow simulations special care is necessary to handle shocks and discontinuities in a stable manner. To meet this, we utilize the shock handling strategy using the GP-based smoothness indicators developed in the previous GP work by A. Reyes et al. We demonstrate the efficacy of the GP-AMR method in a series of testsuite problems using the AMReX library, in which the GP-AMR method has been implemented

Gaussian Process Modeling for Upsampling Algorithms With Applications in Computer Vision and Computational Fluid Dynamics

Across a variety of fields, interpolation algorithms have been used to upsample lowresolution or coarse data fields. In this work, novel Gaussian Process based methodsare employed to solve a variety of upsampling problems. Specifically threeapplications are explored: coarse data prolongation in Adaptive Mesh Refinement(AMR) in the field of Computational Fluid Dynamics, accurate document imageupsampling to enhance Optical Character Recognition (OCR) accuracy, and fastand accurate Single Image Super Resolution (SISR). For AMR, a new, efficient,and “3rd order accurate” algorithm called GP-AMR is presented. Next, a novel,non-zero mean, windowed GP model is generated to upsample low resolution documentimages to generate a higher OCR accuracy, when compared to the industrystandard. Finally, a hybrid GP convolutional neural network algorithm is used togenerate a computationally efficient and high quality SISR model

Développement d’une méthode de simulation d’écoulements à bulles et à gouttes.

L’un des problèmes majeurs rencontré dans la simulation directe des écoulements diphasiques concerne le suivi précis des interfaces au cours du temps et la gestion des changements de topologie (déformation, rupture, coalescence). Les méthodes de simulation où les interfaces évoluent librement sur un maillage fixe permettent de décrire efficacement ces changements de topologie. Le travail présenté ici concerne le développement d’un outil numérique efficace permettant de décrire des écoulements diphasiques dont les rapports de densité et de viscosité peuvent être grands et prenant en compte les effets capillaires. Les applications visées concernent aussi bien les problèmes posés par le génie des procédés que par la propulsion ou les échanges océan-atmosphère. Le transport des interfaces est assuré par une méthode de capture de front sans reconstruction. On montre que la version de base de cette méthode épaissit les zones interfaciales dans les régions où l’écoulement est fortement étiré et on propose une technique de modification de la vitesse dans ces zones qui permet de s’affranchir du problème. Le nouvel algorithme permet aux interfaces de se déplacer librement tout en conservant une épaisseur numérique constante d’environ trois cellules de calcul. Il est testé sur de nombreux écoulements non uniformes et la méthode globale est validée sur différents problèmes de complexité croissante par comparaison avec d’autres résultats théoriques, expérimentaux ou numériques de référence. L’outil numérique ainsi amélioré permet l’étude détaillée de plusieurs aspects de la dynamique des écoulements à bulles et à gouttes intervenant sur une gamme d’échelles de longueur allant de quelques dizaines de microns (mélange dans une goutte en micro-canal) à quelques centimètres (interactions au sein d’un nuage de bulles) dans des configurations axisymétriques ou pleinement tridimensionnelles. Les résultats concernant la microfluidique sont comparés à des expériences très récentes. Une série importante de simulations concernant la montée d’une bulle à travers un liquide, voire deux liquides superposés, est ensuite discutée et validée par rapport aux résultats de la littérature. Enfin des simulations tridimensionnelles de la dynamique d’une suspension comprenant jusqu’à 27 bulles ont été réalisées et analysées. Ces simulations permettent notamment de mettre en évidence l’influence du nombre de Reynolds des bulles sur l’intensité des fluctuations de vitesse qu’elles induisent dans le liquide. ABSTRACT One of the major technical issues in the area of direct simulation of incompressible two-phase flows is to deal with an accurate tracking of interfaces as time proceeds and with changes in interface shape and topology. Numerical methods where interfaces freely evolve on a fixed grid have proved to be efficient for treating such complex phenomena. This work deals with the development of an interface-capturing method aimed at computing three-dimensional incompressible two-phase flows that may involve high density and viscosity ratios and capillary effects. The applications we have in mind concerns chemical engineering as well as environmental problems. We use a front-capturing method to advance the interface but do not perform any explicit reconstruction. We show that the base version of this method results in a smearing of the fronts in regions where the flow undergoes a stray stretching. We propose an improved technique in which the local velocity field within the fronts is modified and the above problem is fixed. This algorithm allows the interfaces to deform properly while maintaining the numerical thickness of the transition region within three computational cells. The overall transport algorithm is tested in several nonuniform flows and the whole numerical method is validated by comparing computational results with analytical solutions and available experimental or numerical data. A detailed study of several aspects of the dynamics of two- and three-phase flows, such as drops in microchannels or hydrodynamic interactions in a bubble swarm, is then performed in both axisymmetric and three-dimensional configurations. The results concerning microfluidics are compared with very recent experiments. A large series of computations concerned with the rise of a single bubble through a liquid or through two superimposed liquids is then discussed and validated against existing data. Finally, threedimensional computation of the dynamics of a bubbly suspension involving up to 27 bubbles are carried out and analyzed. Among others things these simulations allow us to enlighten the influence of the bubbles Reynolds number on the velocity fluctuations induced in the liquid

Schémas numérique d'ordre élevé en temps et en espace pour l'équation des ondes du premier ordre. Application à la Reverse Time Migration.

L imagerie du sous-sol par équations d onde est une application de l ingénierie pétrolière qui mobilise des ressources de calcul très importantes. On dispose aujourd hui de calculateurs puissants qui rendent accessible l imagerie de régions complexes mais des progrès sont encore nécessaires pour réduire les coûts de calcul et améliorer la qualité des simulations. Les méthodes utilisées aujourd hui ne permettent toujours pas d imager correctement des régions très hétérogènes 3D parce qu elles sont trop coûteuses et /ou pas assez précises. Les méthodes d éléments finis sont reconnues pour leur efficacité à produire des simulations de qualité dans des milieux hétérogènes. Dans cette thèse, on a fait le choix d utiliser une méthode de Galerkine discontinue (DG) d ordre élevé à flux centrés pour résoudre l équation des ondes acoustiques et on développe un schéma d ordre élevé pour l intégration en temps qui peut se coupler avec la technique de discrétisation en espace, sans générer des coûts de calcul plus élevés qu avec le schéma d ordre deux Leap-Frog qui est le plus couramment employé. Le nouveau schéma est comparé au schéma d ordre élevé ADER qui s avère plus coûteux car il requiert un plus grand nombre d opérations pour un niveau de précision fixé. De plus, le schéma ADER utilise plus de mémoire, ce qui joue aussi en faveur du nouveau schéma car la production d images du sous-sol consomme beaucoup de mémoire et justifie de développer des méthodes numériques qui utilisent la mémoire au minimum. On analyse également la précision des deux schémas intégrés dans un code industriel et appliqués à des cas test réalistes. On met en évidence des phénomènes de pollution numériques liés à la mise en oeuvre d'une source ponctuelle dans le schéma DG et on montre qu'on peut éliminer ces ondes parasites en introduisant un terme de pénalisation non dissipatif dans la formulation DG. On finit cette thèse en discutant les difficultés engendrées par l'utilisation de schémas numériques dans un contexte industriel, et en particulier l'effet des calculs en simple précision.Oil engineering uses a wide variety of technologies including imaging wave equation which involves very large computing resources. Very powerful computers are now available that make imaging of complex areas possible, but further progress is needed both to reduce the computational cost and improve the simulation accuracy. The current methods still do not allow to image properly heterogeneous 3D regions because they are too expensive and / or not accurate enough. Finite element methods turn out to be efficient for producing good simulations in heterogeneous media. In this thesis, we thus chose to use a high order Discontinuous Galerkin (DG) method based upon centered fluxes to solve the acoustic wave equation and developed a high-order scheme for time integration which can be coupled with the space discretization technique, without generating higher computational cost than the second-order Leap Frog scheme which is the most widely used . The new scheme is compared to the high order ADER scheme which is more expensive because it requires a larger number of computations for a fixed level of accuracy. In addition, the ADER scheme uses more memory, which also works in favor of the new scheme since producing subsurface images consumes lots of memory and justifies the development of low-memory numerical methods. The accuracy of both schemes is then analyzed when they are included in an industrial code and applied to realistic problems. The comparison highlights the phenomena of numerical pollution that occur when injecting a point source in the DG scheme and shows that spurious waves can be eliminated by introducing a non-dissipative penalty term in the DG formulation. This work ends by discussing the difficulties induced by using numerical methods in an industrial framework, and in particular the effect of single precision calculations.PAU-BU Sciences (644452103) / SudocSudocFranceF