Papers by Margarete Domingues
Cartesian CFD Methods for Complex Applications, 2020
Plasma disturbances affect satellites, spacecraft and can cause serious problems to telecommunica... more Plasma disturbances affect satellites, spacecraft and can cause serious problems to telecommunications and sensitive sensor-systems on Earth. Considering the huge scale of the plasma phenomena, data collection at individual locations is not sufficient to cover this entire relevant environment. Therefore, computational plasma modelling has become a significant issue for space sciences, particularly for the near-Earth magnetosphere. However, the simulations of these disturbances present many physical as well as numerical and computational challenges. In this work, we discuss our recent magnetohydrodynamic solver, realised within the MPI-parallel AMROC (Adaptive Mesh Refinement in Object-oriented C ++) framework, in which particular physical models and automatic mesh generation procedures have been implemented. A performance analysis using a selection of significant space applications validates the solvers capabilities and confirms the technical importance of our approach.
Metodos adaptativos tem como objetivo proporcionar solucoes de simulacoes numericas de equacoes d... more Metodos adaptativos tem como objetivo proporcionar solucoes de simulacoes numericas de equacoes diferenciais evolutivas em um tempo computacional baixo, alem de visar uma economia de memoria computacional. Tais caracteristicas sao de interesse das ciencias espaciais, que precisam de m´etodos capazes de resolver problemas robustos em pouco tempo de computacao. [...]
As is well known, the MR technique improves the performance of the standard nite volumes (FV) met... more As is well known, the MR technique improves the performance of the standard nite volumes (FV) methods in space, introducing an adaptive grid which is less rened in the regions where the solution is smoother, instead of using a uniform, regular grid everywhere. In time, an explicit integration either with or without local time-stepping is used. Here, we present an approach, tailored for high performance computing, in which the wavelet-based multiresolution (MR) technique (3) is combined with the block-based adaptive mesh rene- ment (AMR) method (2). In detail, the MR smoothness detector is incorporated as a mesh adaptation criterion in the block-based and fully parallelized mesh renement software AM- ROC (1). Standard hydrodynamics test cases are considered and analysed in terms of total computational time, accuracy and required memory. The accuracy is evaluated by comparing the results of the adaptive computations with those obtained with the corresponding nite volume scheme using a...
Adv. Comput. Math., 2021
Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in... more Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresolution analysis is applied which guarantees the precision of the adaptive computations, while reducing CPU time and memory requirements. Implementation issues of the open source code CARMEN-MHD are discussed. To illustrate its precision and efficiency different benchmark computations including shock-cloud interaction and magnetic reconnection are presented.
Computers & Fluids, 2020
Dynamic mesh adaptation methods require suitable refinement indicators. In the absence of a compr... more Dynamic mesh adaptation methods require suitable refinement indicators. In the absence of a comprehensive error estimation theory, adaptive mesh refinement (AMR) for nonlinear hyperbolic conservation laws, e.g. compressible Euler equations, in practice utilizes mainly heuristic smoothness indicators like combinations of scaled gradient criteria. As an alternative, we describe in detail an easy to implement and computationally inexpensive criterion built on a two-level wavelet transform that applies projection and prediction operators from multiresolution analysis. The core idea is the use of the amplitude of the wavelet coefficients as smoothness indicator, as it can be related to the local regularity of the solution. Implemented within the fully parallelized and structured adaptive mesh refinement (SAMR) software system AMROC (Adaptive Mesh Refinement in Object-oriented C++), the proposed criterion is tested and comprehensively compared to results obtained by applying the scaled gradient approach. A rigorous quantification technique in terms of numerical adaptation error versus used finite volume cells is developed and applied to study typical two-and three-dimensional problems from compressible gas dynamics. It is found that the proposed multiresolution approach is considerably more efficient and also identifiesbesides discontinuous shock and contact waves-in particular smooth rarefaction waves and their interaction as well as small-scale disturbances much more reliably. Aside from pathological cases consisting solely of planar shock waves, the majority of realistic cases show reductions in the number of used finite volume cells between 20 to 40%, while the numerical error remains basically unaltered.
Computers & Fluids, 2019
Computational magnetohydrodynamics (MHD) for space physics has become an essential area in unders... more Computational magnetohydrodynamics (MHD) for space physics has become an essential area in understanding the multiscale dynamics of geophysical and astrophysical plasma processes, partially motivated by the lack of space data. Full MHD simulations are typically very demanding and may require substantial computational efforts. In particular, computational space-weather forecasting is an essential long-term goal in this area, motivated for instance by the needs of modern satellite communication technology. We present a new feature of a recently developed compressible two-and three-dimensional MHD solver, which has been successfully implemented into the parallel AMROC (Adaptive Mesh Refinement in Object-oriented C++) framework with improvements concerning the mesh adaptation criteria based on wavelet techniques. The developments are related to computational efficiency while controlling the precision using dynamically adapted meshes in space-time in a fully parallel context.
Journal of Computational Physics, 2019
A space-time fully adaptive multiresolution method for evolutionary non-linear partial differenti... more A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes, endowed with cell average multiresolution analysis for triggering the dynamical grid adaptation. The explicit time scheme features a natural extension of Runge-Kutta methods which allow local time-stepping while guaranteeing accuracy. The use of a compact Runge-Kutta formulation permits further memory reduction. The precision and computational efficiency of the scheme regarding CPU time and memory compression are assessed for problems in one, two and three space dimensions. As application Burgers equation, reaction-diffusion equations and the compressible Euler equations are considered. The numerical results illustrate the efficiency and superiority of the proposed local time-stepping method with respect to the reference computations.
ESAIM: Proceedings and Surveys, 2016
We present an adaptive multiresolution simulation method for computing weakly compressible flow b... more We present an adaptive multiresolution simulation method for computing weakly compressible flow bounded by solid walls of arbitrary shape, using a finite volume (FV) approach coupled with wavelets for grid adaptation. A volume penalization method is employed to compute the flow in the Cartesian geometry and to impose the boundary conditions. A dynamical adaption strategy to advance both the locally refined grid and the flow in time uses biorthogonal wavelet transforms at each time step. We assess the quality and efficiency of the method for a two-dimensional flow in a periodic channel with a suddenly expanded section. The results are compared with a reference flow obtained by a non-adaptive FV simulation on a uniform grid. It is shown that the adaptive method allows for substantial reduction of CPU time and memory, while preserving the time evolution of the velocity field obtained with the non-adaptive simulation. Résumé. Nous développons une méthode de simulation multi-résolution adaptative pour calculer leś ecoulements faiblement compressibles délimités par des parois solides de forme arbitraire, en utilisant des ondelettes et une approcheen volume fini (FV). La pénalisation en volume est employée pour calculer l'écoulement dans une géométrie cartésienne. Une stratégie d'adaptation dynamique pour avancerà la fois la grille raffinée localement et le flux en temps utilise les ondelettes biorthogonalesà chaque pas de temps. Onévalue la qualité et l'efficacité de la méthode pour unécoulementà deux dimensions dans un canal avec une section soudainementélargie, en comparant avec un calcul de référence obtenu par une méthode en FV non adaptatif. Il est montré que la méthode adaptative permet une réduction substantielle du temps de calcul, tout en préservant l'évolution temporelle du champ de vitesse obtenu avec la simulation non adaptatif.
SIAM Journal on Scientific Computing, 2016
We present a detailed comparison between two adaptive numerical approaches to solve partial diffe... more We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations, adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in space with second order shock-capturing and explicit time integration either with or without local time stepping. The two methods are benchmarked for the compressible Euler equations in Cartesian geometry. As test cases a twodimensional Riemann problem, Lax-Liu #6, and a three-dimensional ellipsoidally expanding shock wave have been chosen. We compare and assess their computational efficiency in terms of CPU time and memory requirements. We evaluate the accuracy by comparing the results of the adaptive computations with those obtained with the corresponding FV scheme using a regular fine mesh. We find that both approaches yield similar trends for CPU time compression for increasing number of refinement levels. MR exhibits more efficient memory compression than AMR and shows slightly enhanced convergence; however, a larger absolute overhead is measured for the tested codes.
ESAIM: Proceedings, 2013
Nous présentons une nouvelle méthode de multirésolution adaptative pour la simulation numérique d... more Nous présentons une nouvelle méthode de multirésolution adaptative pour la simulation numérique de la magnétohydrodynamique idéale. Leséquations qui régissent la dynamique, i.e., leś equations d'Euler compressible couplées auxéquations de Maxwell sont discrétisées suivant un schéma de type volumes finis sur un maillage cartésien en deux dimensions. L'adaptativité en espace est obtenue en utilisant une analyse de multirésolution en moyenne de cellule proposée par Harten, qui est une méthode fiable pour le raffinement local du maillage tout en contrôlant l'érreur. La discrétisation temporelle est un schéma de Runge-Kutta qui intègre un contrôle automatique du pas de temps. Pour imposer l'incompressibilié du champs magnétique, une approche par multiplicateur de Lagrange généralisé est utilisée, ici de type parabolique-hyperbolique. Pour illustrer les capacitées de cette méthode, des applicationsà des problèmes de Riemann ontété réalisées. Les coûts en mémoire sont présentés, et la précision de la méthode estévaluée par comparaison avec les solutions exactes du problème.
ESAIM: Proceedings, 2007
The coherent vortex extraction (CVE) decomposes each turbulent flow realization into two orthogon... more The coherent vortex extraction (CVE) decomposes each turbulent flow realization into two orthogonal components: a coherent and a random incoherent flow. They both contribute to all scales in the inertial range, but exhibit different statistical behaviour. The CVE decomposition is based on the nonlinear filtering of the vorticity field projected onto an orthonormal wavelet basis made of compactly supported functions. We decompose a 3D homogeneous isotropic turbulent flow at Taylor microscale Reynolds numbers R λ = 140 computed by a direct numerical simulation (DNS) at resolution N = 256 3. Only 3.7%N wavelet modes correspond to the coherent flow made of vortex tubes, which contributes to 92% of the enstrophy. Another observation is that the coherent flow exhibits in the inertial range the same k −5/3 slope in the energy spectrum and k 1/3 slope in the enstrophy spectrum as the total flow does. The remaining 96.3%N wavelet modes correspond to a random residual flow which is structureless, quasi equipartition of energy and a Gaussian velocity probability distribution function (PDF). We also analyse and visualize the Lamb vector, its divergence and curl and study the contributions coming from the coherent and incoherent components of vorticity and the induced velocity. * The authors thank the CIRM in Marseille for its hospitality and for financial support during the CEMRACS 2005 summerprogram where part of the work was carried out. 'MF and KS' thankfully acknowledge financial support from the ANR project M2TFP.
ESAIM: Proceedings, 2007
A fully adaptive numerical scheme for solving PDEs based on a finite volume discretization with e... more A fully adaptive numerical scheme for solving PDEs based on a finite volume discretization with explicit time discretization is presented. The local grid refinement is triggered by a multiresolution strategy which allows to control the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. For automatic time step control a Runge-Kutta-Fehlberg method is used. A dynamic tree data structure allows memory compression and CPU time reduction. For validation different classical test problems are computed. The gain in memory and CPU time with respect to the finite volume scheme on a regular grid is reported and demonstrates the efficiency of the new method. Résumé. Nous présentons ici une méthode numérique entièrement adaptative pour les EDP, basée sur une discrétisation spatiale en volumes finis et une intégration temporelle explicite de type Runge-Kutta. Une stratégie de type multi-résolution permet d'adapter localement le maillage tout en contrôlant l'erreur d'approximation en espace. Les flux sontévalués sur la grille adaptative uniquement. Une méthode de type Runge-Kutta-Fehlberg est employée afin de choisir automatiquement le pas de temps tout en contrôlant l'erreur d'approximation. Nous proposons en outre une méthode où le pas de temps dépend de l'échelle, afin d'éviter d'utiliser sur tous les niveaux le pas de temps qui garantit la stabilité numérique sur le niveau de grille le plus fin. La structure de données est organisée en arbre graduel, ce qui permet de réduire significativement la place mémoire et le temps de calcul nécessaires. Nous validons ce nouveau schéma numériqueà l'aide de différents cas-tests classiques. Nous estimons le gain en place mémoire et en temps de calcul par rapport au même calcul en volumes finis sur la grille la plus fine, afin de montrer l'efficacité de la méthode. * The authors thank the CIRM in Marseille for its hospitality and for financial support during the CEMRACS 2005 summerprogram where part of the work was carried out. * * M.O. Domingues thankfully acknowledges financial support from the European Union project IHP on 'Breaking Complexity' (contract HPRN-CT 2002-00286).
Notes on Numerical Fluid Mechanics and Multidisciplinary Design
Applications of the wavelet based coherent vortex extraction method are presented for homogeneous... more Applications of the wavelet based coherent vortex extraction method are presented for homogeneous isotropic turbulence for different Reynolds numbers. We also summarize the developed adaptive multiresolution method for evolutionary PDEs. Then we show first fully adaptive computations of 3d mixing layers using Coherent Vortex Simulation. Features like local scale dependent time stepping are also illustrated and examples for one dimensional problems are given. Test cases on complex geometries like the periodic hill flow (with Reynolds numbers up to 37000) and an annular burner with a swirl number of S = 0.6 have been calculated based on the developed wavelet decomposition models. The extensive results presented show the robustness and good accuracy of the adopted wavelet approach for the various flows simulated.
ESAIM: Proceedings, 2009
Nous présentons des simulations adaptatives multirésolution (MR) deséquations d'Euler compressibl... more Nous présentons des simulations adaptatives multirésolution (MR) deséquations d'Euler compressibles bi-dimensionnelles pour un problème de Riemann classique. Les résultats sont comparés en précision et en efficacité-temps CPU et place mémoire-avec ceux obtenus par la méthode volumes finis sur la grille la plus fine. Pour le même cas-test, nous présentons les calculs obtenusà l'aide de la méthode AMR (Adaptive Mesh Refinement) en imposant les mêmes critères de précision. Les résultats ainsi obtenus sont comparés en termes d'effort de calcul et de compression mémoire, en utilisant des pas de temps globaux puis locaux. De ces résultats préliminaires, nous concluons que les techniques multirésolution présentent des gains en termes de temps CPU et de place mémoire supérieursà ceux de la méthode AMR.
ESAIM: Proceedings, 2011
These lecture notes present adaptive multiresolution schemes for evolutionary PDEs in Cartesian g... more These lecture notes present adaptive multiresolution schemes for evolutionary PDEs in Cartesian geometries. The discretization schemes are based either on finite volume or finite difference schemes. The concept of multiresolution analyses, including Harten's approach for point and cell averages, is described in some detail. Then the sparse point representation method is discussed. Different strategies for adaptive time-stepping, like local scale dependent time stepping and time step control, are presented. Numerous numerical examples in one, two and three space dimensions validate the adaptive schemes and illustrate the accuracy and the gain in computational efficiency in terms of CPU time and memory requirements. Another aspect, modeling of turbulent flows using multiresolution decompositions, the so-called Coherent Vortex Simulation approach is also described and examples are given for computations of three-dimensional weakly compressible mixing layers. Most of the material concerning applications to PDEs is assembled and adapted from previous publications [27, 31, 32, 34, 67, 69].
Space-Time Adaptive Multiresolution Techniques for Compressible Euler Equations
The Courant–Friedrichs–Lewy (CFL) Condition, 2013
ABSTRACT
2007 IEEE Antennas and Propagation International Symposium, 2007
Journal of Computational Physics, 2008
We present a fully adaptive numerical scheme for evolutionary PDEs in Cartesian geometry based on... more We present a fully adaptive numerical scheme for evolutionary PDEs in Cartesian geometry based on a second-order finite volume discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. For time discretization we use an explicit Runge-Kutta scheme of second-order with a scale-dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On larger scales, the time step can be increased without violating the stability requirement of the explicit scheme. The implementation uses a dynamic tree data structure. Numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time-stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid. Fully adaptive three-dimensional computations for reaction-diffusion equations illustrate the additional speed-up of the local time stepping for a thermo-diffusive flame instability.
Journal of Computational and Applied Mathematics, 2010
In the Sparse Point Representation (SPR) method the principle is to retain the function data indi... more In the Sparse Point Representation (SPR) method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors by means of an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Illustrating results are presented for 2D Maxwell's equation numerical solutions.
International Journal for Numerical Methods in Engineering, 2009
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Papers by Margarete Domingues