1,161 research outputs found
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
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
Wavelet-based Adaptive Techniques Applied to Turbulent Hypersonic Scramjet Intake Flows
The simulation of hypersonic flows is computationally demanding due to large
gradients of the flow variables caused by strong shock waves and thick boundary
or shear layers. The resolution of those gradients imposes the use of extremely
small cells in the respective regions. Taking turbulence into account
intensives the variation in scales even more. Furthermore, hypersonic flows
have been shown to be extremely grid sensitive. For the simulation of
three-dimensional configurations of engineering applications, this results in a
huge amount of cells and prohibitive computational time. Therefore, modern
adaptive techniques can provide a gain with respect to computational costs and
accuracy, allowing the generation of locally highly resolved flow regions where
they are needed and retaining an otherwise smooth distribution. An h-adaptive
technique based on wavelets is employed for the solution of hypersonic flows.
The compressible Reynolds averaged Navier-Stokes equations are solved using a
differential Reynolds stress turbulence model, well suited to predict
shock-wave-boundary-layer interactions in high enthalpy flows. Two test cases
are considered: a compression corner and a scramjet intake. The compression
corner is a classical test case in hypersonic flow investigations because it
poses a shock-wave-turbulent-boundary-layer interaction problem. The adaptive
procedure is applied to a two-dimensional confguration as validation. The
scramjet intake is firstly computed in two dimensions. Subsequently a
three-dimensional geometry is considered. Both test cases are validated with
experimental data and compared to non-adaptive computations. The results show
that the use of an adaptive technique for hypersonic turbulent flows at high
enthalpy conditions can strongly improve the performance in terms of memory and
CPU time while at the same time maintaining the required accuracy of the
results.Comment: 26 pages, 29 Figures, submitted to AIAA Journa
Task-based adaptive multiresolution for time-space multi-scale reaction-diffusion systems on multi-core architectures
A new solver featuring time-space adaptation and error control has been
recently introduced to tackle the numerical solution of stiff
reaction-diffusion systems. Based on operator splitting, finite volume adaptive
multiresolution and high order time integrators with specific stability
properties for each operator, this strategy yields high computational
efficiency for large multidimensional computations on standard architectures
such as powerful workstations. However, the data structure of the original
implementation, based on trees of pointers, provides limited opportunities for
efficiency enhancements, while posing serious challenges in terms of parallel
programming and load balancing. The present contribution proposes a new
implementation of the whole set of numerical methods including Radau5 and
ROCK4, relying on a fully different data structure together with the use of a
specific library, TBB, for shared-memory, task-based parallelism with
work-stealing. The performance of our implementation is assessed in a series of
test-cases of increasing difficulty in two and three dimensions on multi-core
and many-core architectures, demonstrating high scalability
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
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