741 research outputs found
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
A New Unconditionally Stable Method for Telegraph Equation Based on Associated Hermite Orthogonal Functions
The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkinâs method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods
WAVELET REGULARIZATION OF A FOURIER-GALERKIN METHOD FOR SOLVING THE 2D INCOMPRESSIBLE EULER EQUATIONS
International audienceWe employ a Fourier-Galerkin method to solve the 2D incompressible Euler equations, and study several ways to regularize the solution by wavelet ïŹltering at each timestep. Real-valued orthogonal wavelets and complex-valued wavelets are considered, combined with either linear or non-linear ïŹltering. The results are compared with those obtained via classical viscous and hyperviscous regularization methods. Wavelet regularization using complex-valued wavelets performs as well in terms of L2 convergence rate to the reference solution. The compression rate for homogeneous 2D turbulence is around 3 for this method, suggesting that memory and CPU time could be reduced in an adaptive wavelet computation. Our results also suggest L2 convergence to the reference solution without any regularization, in contrast to what is obtained for the 1D Burgers equation
Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins
In this work we propose an Uncertainty Quantification methodology for
sedimentary basins evolution under mechanical and geochemical compaction
processes, which we model as a coupled, time-dependent, non-linear,
monodimensional (depth-only) system of PDEs with uncertain parameters. While in
previous works (Formaggia et al. 2013, Porta et al., 2014) we assumed a
simplified depositional history with only one material, in this work we
consider multi-layered basins, in which each layer is characterized by a
different material, and hence by different properties. This setting requires
several improvements with respect to our earlier works, both concerning the
deterministic solver and the stochastic discretization. On the deterministic
side, we replace the previous fixed-point iterative solver with a more
efficient Newton solver at each step of the time-discretization. On the
stochastic side, the multi-layered structure gives rise to discontinuities in
the dependence of the state variables on the uncertain parameters, that need an
appropriate treatment for surrogate modeling techniques, such as sparse grids,
to be effective. We propose an innovative methodology to this end which relies
on a change of coordinate system to align the discontinuities of the target
function within the random parameter space. The reference coordinate system is
built upon exploiting physical features of the problem at hand. We employ the
locations of material interfaces, which display a smooth dependence on the
random parameters and are therefore amenable to sparse grid polynomial
approximations. We showcase the capabilities of our numerical methodologies
through two synthetic test cases. In particular, we show that our methodology
reproduces with high accuracy multi-modal probability density functions
displayed by target state variables (e.g., porosity).Comment: 25 pages, 30 figure
MĂ©thode de Galerkin discontinue pour la discrĂ©tisation par ĂlĂ©ments finis des Ă©quations de maxwell pour la modĂ©lisation de problĂšmes dâĂ©lectromagnĂ©tisme en basses frĂ©quences
RĂSUMĂ: Une discrĂ©tisation par Ă©lĂ©ments finis utilisant la mĂ©thode de Galerkin discontinue pour les Ă©quations de Maxwell est proposĂ©e pour modĂ©liser les problĂšmes dâĂ©lectromagnĂ©tisme en basses frĂ©quences. Lâapproximation des Ă©quations de Maxwell, dans le rĂ©gime des basses
frĂ©quences, est directement discrĂ©tisĂ©e avec la mĂ©thode de Galerkin discontinue qui a Ă©tĂ© originellement dĂ©veloppĂ©e pour les problĂšmes hyperboliques. On Ă©tudie, plus prĂ©cisĂ©ment, la modĂ©lisation des problĂšmes sur les supraconducteurs Ă haute tempĂ©rature afin dâĂ©valuer
la robustesse de la stratĂ©gie numĂ©rique proposĂ©e. Une analyse dimensionnelle du systĂšme dâĂ©quations aux dĂ©rivĂ©es partielles dâordre un, ainsi quâun modĂšle pour les milieux ambiants ayant une conductivitĂ© trĂšs faible sont aussi proposĂ©s. Un problĂšme ayant une solution manufacturĂ©e
et la propagation dâun front magnĂ©tique sont Ă©tudiĂ©s afin de vĂ©rifier la mĂ©thodologie numĂ©rique proposĂ©e. Lâinduction dâun courant Ă©lectrique dans un Ă©chantillon et dans des
cùbles électriques supraconducteurs à configuration complexe est étudiée afin de valider le modÚle mathématique. De plus, une comparaison sur la capture des forts gradients de la
densitĂ© de courant et sur la robustesse du schĂ©ma de points-fixes est faite entre la stratĂ©gie numĂ©rique proposĂ©e et lâapproche numĂ©rique populaire au sein de la communautĂ© des ingĂ©nieurs Ă©lectriques en utilisant les problĂšmes sur les supraconducteurs Ă haute tempĂ©rature.----------ABSTRACT: The discretization of Maxwellâs equations using the discontinuous Galerkin finite element method is proposed for modeling electromagnetism problems in low-frequency regime. The low-frequency approximation to Maxwellâs equations is directly discretized using the discontinuous Galerkin method that was first designed for hyperbolic problems. The modeling of high-temperature superconductors is particularly studied to assess the robustness of the proposed numerical strategy. A dimensional analysis of Maxwellâs equations in low-frequency regime is proposed as well as a model for a medium with very low conductivity, such as air medium. A problem with a manufactured solution and the magnetic front problem are used to verify the proposed numerical strategy. The magnetization of superconducting bulks and wires with a complex structure is used to validate the mathematical model. For hightemperature superconductors modeling, the capture of the sharp gradients of the current density and the robustness of the fixed-point method are studied. The proposed approach is also compared with the popular numerical strategy among the electrical engineering community
- âŠ