353 research outputs found
Adaptive multiresolution computations applied to detonations
A space-time adaptive method is presented for the reactive Euler equations
describing chemically reacting gas flow where a two species model is used for
the chemistry. The governing equations are discretized with a finite volume
method and dynamic space adaptivity is introduced using multiresolution
analysis. A time splitting method of Strang is applied to be able to consider
stiff problems while keeping the method explicit. For time adaptivity an
improved Runge--Kutta--Fehlberg scheme is used. Applications deal with
detonation problems in one and two space dimensions. A comparison of the
adaptive scheme with reference computations on a regular grid allow to assess
the accuracy and the computational efficiency, in terms of CPU time and memory
requirements.Comment: Zeitschrift f\"ur Physicalische Chemie, accepte
Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems
We present a fully adaptive multiresolution scheme for spatially
two-dimensional, possibly degenerate reaction-diffusion systems, focusing on
combustion models and models of pattern formation and chemotaxis in
mathematical biology. Solutions of these equations in these applications
exhibit steep gradients, and in the degenerate case, sharp fronts and
discontinuities. The multiresolution scheme is based on finite volume
discretizations with explicit time stepping. The multiresolution representation
of the solution is stored in a graded tree. By a thresholding procedure, namely
the elimination of leaves that are smaller than a threshold value, substantial
data compression and CPU time reduction is attained. The threshold value is
chosen optimally, in the sense that the total error of the adaptive scheme is
of the same slope as that of the reference finite volume scheme. Since chemical
reactions involve a large range of temporal scales, but are spatially well
localized (especially in the combustion model), a locally varying adaptive time
stepping strategy is applied. It turns out that local time stepping accelerates
the adaptive multiresolution method by a factor of two, while the error remains
controlled.Comment: 27 pages, 14 figure
A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
This work deals with the numerical solution of the monodomain and bidomain
models of electrical activity of myocardial tissue. The bidomain model is a
system consisting of a possibly degenerate parabolic PDE coupled with an
elliptic PDE for the transmembrane and extracellular potentials, respectively.
This system of two scalar PDEs is supplemented by a time-dependent ODE modeling
the evolution of the so-called gating variable. In the simpler sub-case of the
monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple
models for the membrane and ionic currents are considered, the
Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical
solutions of the bidomain and monodomain models exhibit wavefronts with steep
gradients, we propose a finite volume scheme enriched by a fully adaptive
multiresolution method, whose basic purpose is to concentrate computational
effort on zones of strong variation of the solution. Time adaptivity is
achieved by two alternative devices, namely locally varying time stepping and a
Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical
examples demonstrates thatthese methods are efficient and sufficiently accurate
to simulate the electrical activity in myocardial tissue with affordable
effort. In addition, an optimalthreshold for discarding non-significant
information in the multiresolution representation of the solution is derived,
and the numerical efficiency and accuracy of the method is measured in terms of
CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure
Multiresolution strategies for the numerical solution of optimal control problems
Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme.
The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a nonlinear programming (NLP) problem that is solved using standard NLP codes. The novelty of the proposed approach hinges on the automatic calculation of a suitable, nonuniform grid over which the NLP problem is solved, which tends to increase numerical efficiency and robustness. Control and/or state constraints are handled with ease, and without any additional computational complexity. The proposed algorithm is based on a simple and intuitive method to balance several conflicting objectives, such as accuracy of the solution, convergence, and speed of the computations. The benefits of the proposed algorithm over uniform grid implementations are demonstrated with the help of several nontrivial examples. Furthermore, two sequential multiresolution trajectory optimization algorithms for solving problems with moving targets and/or dynamically changing environments have been developed.Ph.D.Committee Chair: Tsiotras, Panagiotis; Committee Member: Calise, Anthony J.; Committee Member: Egerstedt, Magnus; Committee Member: Prasad, J. V. R.; Committee Member: Russell, Ryan P.; Committee Member: Zhou, Hao-Mi
An open and parallel multiresolution framework using block-based adaptive grids
A numerical approach for solving evolutionary partial differential equations
in two and three space dimensions on block-based adaptive grids is presented.
The numerical discretization is based on high-order, central finite-differences
and explicit time integration. Grid refinement and coarsening are triggered by
multiresolution analysis, i.e. thresholding of wavelet coefficients, which
allow controlling the precision of the adaptive approximation of the solution
with respect to uniform grid computations. The implementation of the scheme is
fully parallel using MPI with a hybrid data structure. Load balancing relies on
space filling curves techniques. Validation tests for 2D advection equations
allow to assess the precision and performance of the developed code.
Computations of the compressible Navier-Stokes equations for a temporally
developing 2D mixing layer illustrate the properties of the code for nonlinear
multi-scale problems. The code is open source
A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations
This paper presents a new resolution strategy for multi-scale streamer
discharge simulations based on a second order time adaptive integration and
space adaptive multiresolution. A classical fluid model is used to describe
plasma discharges, considering drift-diffusion equations and the computation of
electric field. The proposed numerical method provides a time-space accuracy
control of the solution, and thus, an effective accurate resolution independent
of the fastest physical time scale. An important improvement of the
computational efficiency is achieved whenever the required time steps go beyond
standard stability constraints associated with mesh size or source time scales
for the resolution of the drift-diffusion equations, whereas the stability
constraint related to the dielectric relaxation time scale is respected but
with a second order precision. Numerical illustrations show that the strategy
can be efficiently applied to simulate the propagation of highly nonlinear
ionizing waves as streamer discharges, as well as highly multi-scale nanosecond
repetitively pulsed discharges, describing consistently a broad spectrum of
space and time scales as well as different physical scenarios for consecutive
discharge/post-discharge phases, out of reach of standard non-adaptive methods.Comment: Support of Ecole Centrale Paris is gratefully acknowledged for
several month stay of Z. Bonaventura at Laboratory EM2C as visiting
Professor. Authors express special thanks to Christian Tenaud (LIMSI-CNRS)
for providing the basis of the multiresolution kernel of MR CHORUS, code
developed for compressible Navier-Stokes equations (D\'eclaration d'Invention
DI 03760-01). Accepted for publication; Journal of Computational Physics
(2011) 1-2
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