Reconstruction based error detection for robust approximation of partial differential equations

In this work we present a framework for the construction of robust a posteriori estimates for classes of finite difference schemes. We are motivated by the relative lack of such frameworks compared to existing ones for other numerical discretisation methods, such as finite elements and finite volumes. The framework we propose is based on the use of reconstructions, which are obtained by post-processing the finite difference solution. The post-processed object is a key ingredient in obtaining a posteriori bounds using the relevant stability framework of the problem. The resulting bounds are fully computable and allow us to establish a posteriori error control over the problem at hand. In the first part of the thesis we motivate and investigate the behaviour of our framework using model ODE, elliptic and hyperbolic problems. We use our framework to obtain reconstructions which are used to compute a posteriori error estimates. We validate the numerical behaviour of these estimates using solutions of varying regularity. In the second part of the thesis we focus on hyperbolic conservation laws in one spatial dimension and we deal with scalar problems as well as systems. Hyperbolic conservation laws are widely used in the modelling of physical phenomena. The numerical modelling of conservation laws, which arises due to the frequent lack of explicit solutions, is challenging, largely due to the complex behaviour these problems exhibit, such as shock formation even with smooth initial conditions. In this setting, we present a framework which is applicable to general non-linear conservation laws. We investigate its numerical behaviour and showcase our results by using popular finite difference discretisations for a range of problems. We demonstrate that the the framework can produce optimal estimates, capable of tracking features of interest and act as refinement/coarsening indicators

La méthode MOOD Multi-dimensional Optimal Order Detection : la première approche a posteriori aux méthodes volumes finis d'ordre très élevé

Nous introduisons et développons dans cette thèse un nouveau type de méthodes Volumes Finis d'ordre très élevé pour les systèmes hyperboliques de lois de conservations. Appelée MOOD pour Multidimensional Optimal Order Detection, elle permet de réaliser des simulations très précises en dimensions deux et trois sur maillages non-structurés. La conception d'une telle méthode est rendue délicate par l'apparition de singularités dans la solution (chocs, discontinuités de contact) pour lesquelles des phenomènes parasites (oscillations, création de valeurs non physiques...) sont générés par l'approximation d'ordre élevé. L'originalité de cette thèse réside dans le traitement de ces problèmes. A l'opposé des méthodes classiques qui essaient de contrôler ces phénomènes indésirables par une limitation a priori, nous proposons une approche de traitement a posteriori basée sur une décrémentation locale de l'ordre du schéma. Nous montrons en particulier que ce concept permet très simplement d'obtenir des propriétés qui sont habituellement difficiles à prouver dans le cadre multi-dimensionel non-structuré (préservation de la positité par exemple). La robustesse et la qualité de la méthode MOOD ont été prouvées sur de nombreux tests numériques en 2D et 3D. Une amélioration significative des ressources informatiques (CPU et stockage mémoire) nécessaires à l'obtention de résultats équivalents aux méthodes actuelles a été démontrée.We introduce and develop in this thesis a new type of very high-order Finite Volume methods for hyperbolic systems of conservation laws. This method, named MOOD for Multidimensional Optimal Order Detection, provides very accurate simulations for two- and three-dimensional unstructured meshes. The design of such a method is made delicate by the emergence of solution singularities (shocks, contact discontinuities) for which spurious phenomena (oscillations, non-physical values creation, etc.) are generated by the high-order approximation. The originality of this work lies in a new treatment for theses problems. Contrary to classical methods which try to control such undesirable phenomena through an a priori limitation, we propose an a posteriori treatment approach based on a local scheme order decrementing. In particular, we show that this concept easily provides properties that are usually difficult to prove in a multidimensional unstructured framework (positivity-preserving for instance). The robustness and quality of the MOOD method have been numerically proved through numerous test cases in 2D and 3D, and a significant reduction of computational resources (CPU and memory storage) needed to get state-of-the-art results has been shown

Edge Detection by Adaptive Splitting II. The Three-Dimensional Case

In Llanas and Lantarón, J. Sci. Comput. 46, 485–518 (2011) we proposed an algorithm (EDAS-d) to approximate the jump discontinuity set of functions defined on subsets of ℝ d . This procedure is based on adaptive splitting of the domain of the function guided by the value of an average integral. The above study was limited to the 1D and 2D versions of the algorithm. In this paper we address the three-dimensional problem. We prove an integral inequality (in the case d=3) which constitutes the basis of EDAS-3. We have performed detailed computational experiments demonstrating effective edge detection in 3D function models with different interface topologies. EDAS-1 and EDAS-2 appealing properties are extensible to the 3D cas

Coupling of hybridisable discontinuous Galerkin and finite volumes for transient compressible flows

Fast, high-fidelity solution workflows for transient flow phenomena is an important challenge in the computational fluid dynamics (CFD) community. Current low-order methodologies suffer from large dissipation and dispersion errors and require large mesh sizes for unsteady flow simulations. Recently, on the other hand, high-order methods have gained popularity offering high solution accuracy. But they suffer from the lack of robust, curvilinear mesh generators.A novel methodology that combines the advantages of the classical vertex-centred finite volume (FV) method and high-order hybridisable discontinuous Galerkin (HDG) method is presented for the simulation of transient inviscid compressible flows. The resulting method is capable of simulating the transient effects on coarse, unstructured meshes that are suitable to perform steady simulations with traditional low-order methods. In the vicinity of the aerodynamic shapes, FVs are used whereas in regions where the size of the element is too large for finite volumes to provide an accurate answer, the high-order HDG approach is employed with a non-uniform degree of approximation. The proposed method circumvents the need to produce tailored meshes for transient simulations, as required in a low-order context, and also the need to produce high-order curvilinear meshes, as required by high-order methods.FV and HDG methods for compressible inviscid flows with an implicit time-stepping method and capable of handling flow discontinuities is developed. A two-way coupling of the methods in a monolithic manner was achieved by the consistent application of the so-called transmission conditions at the FV-HDG interface. Numerical tests highlight the optimal convergence properties of the coupled HDG-FV scheme. Numeri-cal examples demonstrate the potential and suitability of the developed methodology for unsteady 2D and 3D flows in the context of simulating the wind gust effect on aerodynamic shapes

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

ICASE/LaRC Workshop on Adaptive Grid Methods

Solution-adaptive grid techniques are essential to the attainment of practical, user friendly, computational fluid dynamics (CFD) applications. In this three-day workshop, experts gathered together to describe state-of-the-art methods in solution-adaptive grid refinement, analysis, and implementation; to assess the current practice; and to discuss future needs and directions for research. This was accomplished through a series of invited and contributed papers. The workshop focused on a set of two-dimensional test cases designed by the organizers to aid in assessing the current state of development of adaptive grid technology. In addition, a panel of experts from universities, industry, and government research laboratories discussed their views of needs and future directions in this field

Advanced numerical approaches in the dynamics of relativistic flows

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A Mixed Hybrid Finite Volumes Solver for Robust Primal and Adjoint CFD

PhDIn the context of gradient-based numerical optimisation, the adjoint method is an e cient way of computing the gradient of the cost function at a computational cost independent of the number of design parameters, which makes it a captivating option for industrial CFD applications involving costly primal solves. The method is however a ected by instabilities, some of which are inherited from the primal solver, notably if the latter does not fully converge. The present work is an attempt at curbing primal solver limitations with the goal of indirectly alleviating adjoint robustness issues. To that end, a novel discretisation scheme for the steady-state incompressible Navier- Stokes problem is proposed: Mixed Hybrid Finite Volumes (MHFV). The scheme draws inspiration from the family of Mimetic Finite Di erences and Mixed Virtual Elements strategies, rid of some limitations and numerical artefacts typical of classical Finite Volumes which may hinder convergence properties. Derivation of MHFV operators is illustrated and each scheme is validated via manufactured solutions: rst for pure anisotropic di usion problems, then convection-di usion-reaction and nally Navier-Stokes. Traditional and novel Navier-Stokes solution algorithms are also investigated, adapted to MHFV and compared in terms of performance. The attention is then turned to the discrete adjoint Navier-Stokes system, which is assembled in an automated way following the principles of Equational Di erentiation, i.e. the di erentiation of the primal discrete equations themselves rather than the algorithm used to solve them. Practical/computational aspects of the assembly are discussed, then the adjoint gradient is validated and a few solution algorithms for the MHFV adjoint Navier-Stokes are proposed and tested. Finally, two examples of full shape optimisation procedures on internal ow test cases (S-bend and U-bend) are reported.European Union's Seventh Framework Programme grant agreement number 317006

A hermite radial basis functions control volume numerical method to simulate transport problems

This thesis presents a Control Volume (CV) method for transient transport problems where the cell surface fluxes are reconstructed using local interpolation functions that besides interpolating the nodal values of the field variable, also satisfies the governing equation at some auxiliary points in the interpolation stencils. The interpolation function relies on a Hermitian Radial Basis Function (HRBF) mesh less collocation approach to find the solution of auxiliary local boundary/initial value problems, which are solved using the same time integration scheme adopted to update the global control volume solution. By the use of interpolation functions that approximate the governing equation, a form of analytical upwinding scheme is achieved without the need of using predefined interpolation stencils according to the magnitude and direction of the local advective velocity. In this way, the interpolation formula retains the desired information about the advective velocity field, allowing the use of centrally defined stencils even in the case of advective dominant problems. This new CV approach, which is referred to as the CV-HRBF method, is applied to a series of transport problems characterised by high Peclet number. This method is also more flexible than the classical CV formulations because the boundary conditions are explicitly imposed in the interpolation formula, without the need for artificial schemes (e.g. utilising dummy cells). The flexibility of the local meshless character of the CVHRBF is shown in the modelling of the saturated zone of the unconfined aquifer where a mesh adapting algorithm is needed to track the phreatic surface (moving boundary). Due to the use of a local RBF interpolation, the dynamic boundary condition can be applied in an arbitrary number of points on the phreatic surface, independently from the mesh element. The robustness of the Hermite interpolation is exploited to formulate a non-overlapping non-iterative multi-domain scheme where physical matching conditions are satisfied locally, i.e. imposing the continuity of the function and flux at the sub-domain interface