516 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
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
Adaptive sparse grid discontinuous Galerkin method: review and software implementation
This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG)
method for computing high dimensional partial differential equations (PDEs) and
its software implementation. The C\texttt{++} software package called AdaM-DG,
implementing the aSG-DG method, is available on Github at
\url{https://github.com/JuntaoHuang/adaptive-multiresolution-DG}. The package
is capable of treating a large class of high dimensional linear and nonlinear
PDEs. We review the essential components of the algorithm and the functionality
of the software, including the multiwavelets used, assembling of bilinear
operators, fast matrix-vector product for data with hierarchical structures. We
further demonstrate the performance of the package by reporting numerical error
and CPU cost for several benchmark test, including linear transport equations,
wave equations and Hamilton-Jacobi equations
Hierarchical bases for non-hierarchic 3Dtriangular meshes
We describe a novel basis of hierarchical, multiscale functions that are linear combinations of standard Rao-Wilton- Glisson (RWG) functions. When the basis is used for discretizing the electric field integral equation (EFIE) for PEC objects it gives rise to a linear system immune from low-frequency breakdown, and well conditioned for dense meshes. The proposed scheme can be applied to any mesh with triangular facets, and therefore it can be used as if it were an algebraic preconditioner. The properties of the new system are confirmed by numerical results that show fast convergence rates of iterative solvers, significantly better than those for the loop-tree basis. As a byproduct of the basis generation, a generalization of the RWG functions to nonsimplex cells is introduced
Performance study of the multiwavelet discontinuous Galerkin approach for solving the GreenâNaghdi equations
This paper presents a multiresolution discontinuous Galerkin scheme for the adaptive solution of Boussinesqâtype equations. The model combines multiwaveletâbased grid adaptation with a discontinuous Galerkin (DG) solver based on the system of fully nonlinear and weakly dispersive GreenâNaghdi (GN) equations. The key feature of the adaptation procedure is to conduct a multiresolution analysis using multiwavelets on a hierarchy of nested grids to improve the efficiency of the reference DG scheme on a uniform grid by computing on a locally refined adapted grid. This way the local resolution level will be determined by manipulating multiwavelet coefficients controlled by a single userâdefined threshold value. The proposed adaptive multiwavelet discontinuous Galerkin solver for GN equations (MWDGâGN) is assessed using several benchmark problems related to wave propagation and transformation in nearshore areas. The numerical results demonstrate that the proposed scheme retains the accuracy of the reference scheme, while significantly reducing the computational cost
Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux
A fully adaptive finite volume multiresolution scheme for one-dimensional
strongly degenerate parabolic equations with discontinuous flux is presented.
The numerical scheme is based on a finite volume discretization using the
Engquist--Osher approximation for the flux and explicit time--stepping. An
adaptivemultiresolution scheme with cell averages is then used to speed up CPU
time and meet memory requirements. A particular feature of our scheme is the
storage of the multiresolution representation of the solution in a dynamic
graded tree, for the sake of data compression and to facilitate navigation.
Applications to traffic flow with driver reaction and a clarifier--thickener
model illustrate the efficiency of this method
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