Fully Adaptive Multiscale Schemes for Conservation Laws Employing Locally Varying Time Stepping

Müller, Siegfried; Stiriba, Youssef

doi:10.1007/s10915-006-9102-z

Cited by 79 publications

(80 citation statements)

References 38 publications

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“…Work currently in preparation includes other applications of interest that maintain the same essential structure: strong degeneracy of the diffusive terms, but for systems and higher space dimensions (by means of tensor product strategies). Other possible extensions include the use of penalisation methods for complex domains and schemes with local time adaptivity [22,41]. As an observation of a practical nature for readers who may wish to implement the method presented herein, we mention that in order to obtain memory compression, not only the tree data structure is essential, but also the way of navigating inside the data structure, as proposed in [45].…”

Section: Discussionmentioning

confidence: 99%

“…Though the version of the multiresolution method of [13] is effective for (1.1), it does not provide memory savings. In contrast to [13], the method presented herein does provide significant memory savings, since the multiresolution representation of the solution is stored in a graded tree [17,22,40,41,45], whose leaves are the finite volumes for which the numerical divergence is computed. This means that the numerical flux is actually evaluated on the borders of these finite volumes.…”

Section: Introductionmentioning

confidence: 99%

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Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Bürger¹,

Ruiz²,

Schneider³

et al. 2008

ESAIM: M2AN

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Abstract. We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L 1 contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties.Mathematics Subject Classification. 35L65, 35R05, 65M06, 76T20.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Bürger¹,

Ruiz²,

Schneider³

et al. 2008

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…We mention, for example, [8,29], where it is shown that computational savings in memory requirements can also be obtained by incorporating adequate data structures in the numerical code. This strategy can turn a wavelet-based adaptive scheme into a true adaptive mesh-refinement code (see [25]). In the present paper, the multilevel technique reduces the effective CPU time associated to the computation of the numerical divergence (which is the main time consuming part of the full algorithm) by an average factor of 5.5 for uniform grids with 1536 2 points, of 4 for 1024 2 , and of 3 for 512 2 grid points.…”

Section: Discretization Of the Inviscid Fluxesmentioning

confidence: 99%

A high-resolution penalization method for large Mach number flows in the presence of obstacles

2009

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A penalization method is applied to model the interaction of large Mach number compressible flows with obstacles. A supplementary term is added to the compressible Navier-Stokes system, seeking to simulate the effect of the Brinkmanpenalization technique used in incompressible flow simulations including obstacles. We present a computational study comparing numerical results obtained with this method to theoretical results and to simulations with Fluent software. Our work indicates that this technique can be very promising in applications to complex flows.

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“…Nevertheless, the CPU-time can be significantly reduced using the local time stepping algorithm, see e.g. [2,16,25]. In recent years, such numerical schemes have been widely developed for equations which arise in many fluid flows: traffic flows, multi-phase flows, multi-layer flows, etc.…”

Section: Introductionmentioning

confidence: 99%

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Ersoy¹,

Golay²,

Yushchenko³

2013

Open Mathematics

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We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod's shock tube problem, Lax's shock tube problem and the Shu-Osher test problem. MSC:

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Fully Adaptive Multiscale Schemes for Conservation Laws Employing Locally Varying Time Stepping

Cited by 79 publications

References 38 publications

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

A high-resolution penalization method for large Mach number flows in the presence of obstacles

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

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