Solution of shallow water equations using fully adaptive multiscale schemes

Lamby, Philipp; Müller, Siegfried; Stiriba, Youssef

doi:10.1002/fld.1004

Cited by 35 publications

(28 citation statements)

References 31 publications

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“…From Fig. 15 we can observe that if we incorporate the local time stepping strategy, a substantial gain (a factor slightly lower than 2, which is consistent with the results of [23]) is obtained in speed-up rate when comparing with a multiresolution calculation using global time stepping. The errors are computed from a reference solution in a grid with The compression rate η for both methods is lower than in the previous examples, which could be explained by the complexity and density of the spatial patterns in this particular example.…”

Section: Example 6: Chemotaxis-growth Systemsupporting

confidence: 90%

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction–diffusion systems

Bendahmane

Bürger²,

Ruiz–Baier

et al. 2009

Applied Numerical Mathematics

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MSC: 35K65 35L65 35R05 65M06 76T20 92C17 Keywords: Degenerate parabolic equation Adaptive multiresolution scheme Pattern formation Finite volume schemes Chemotaxis Keller-Segel systems Flame balls interaction Locally varying time steppingSpatially two-dimensional, possibly degenerate reaction-diffusion systems, with a focus on models of combustion, pattern formation and chemotaxis, are solved by a fully adaptive multiresolution scheme. Solutions of these equations exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort on zones of strong variation. 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 ("quadtree"), whose leaves are the non-uniform finite volumes on whose borders the numerical divergence is evaluated. By a thresholding procedure, namely the elimination of leaves with solution values that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen such that the total error of the adaptive scheme is of the same order 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. For scalar equations, this strategy has the advantage that consistence with a CFL condition is always enforced. Numerical experiments with five different scenarios, in part with local time stepping, illustrate the effectiveness of the adaptive multiresolution method. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.

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Section: Example 6: Chemotaxis-growth Systemsupporting

confidence: 90%

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction–diffusion systems

Bendahmane

Bürger²,

Ruiz–Baier

et al. 2009

Applied Numerical Mathematics

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“…The wave-topography interaction has produced a highly complex wave pattern as indicated by the 3D water surface as well as the depth contours. The results compare very well with those presented by Tang [34] and Lamby et al [35]. Similar to those produced by Lamby et al [35] who used a fine adapted mesh with 261 720 cells, the fronts of the incident bore together with the reflected shocks are sharply predicted inside the refined domain.…”

Section: Bore Wave Past a Dipsupporting

confidence: 86%

“…The results compare very well with those presented by Tang [34] and Lamby et al [35]. Similar to those produced by Lamby et al [35] who used a fine adapted mesh with 261 720 cells, the fronts of the incident bore together with the reflected shocks are sharply predicted inside the refined domain. However, it is interesting to observe that, outside the high-resolution mesh, diffusive bore fronts are computed.…”

Section: Bore Wave Past a Dipsupporting

confidence: 86%

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A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

Liang¹

2011

Numerical Methods in Fluids

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SUMMARYIn large-scale shallow flow simulations, local high-resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non-uniform Cartesian grid system as an alternative to the existing approaches to provide local high-resolution mesh. On generating a structured but non-uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second-order finite volume Godunov-type scheme in a similar way as that on a uniform grid.

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“…where G HI i+ 1 2 is given by (24) (it leads to a high order scheme), and G LO i+ 1 2 is chosen so that its inclusion in (23) leads to a monotone (non-oscillatory) low order scheme. The following choice is considered in [27] G LO i+ )(g n i+1 − g n i )), (27) where sgn(x) is the signum function.…”

Section: The Tvdb Schemementioning

confidence: 99%

Cost-effective Multiresolution schemes for Shock Computations

2009

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Abstract. Harten's Multiresolution framework has provided a fruitful environment for the development of adaptive codes for hyperbolic PDEs. The so-called cost-effective alternative [4,8,21] seeks to achieve savings in the computational cost of the underlying numerical technique, but not in the overall memory requirements of the code. Since the data structure of the basic algorithm does not need to be modified, it provides a set of tools that can be easily implemented into existing codes and that can be very useful in order to speed up the numerical simulations involved in the testing process that is associated to the development of new numerical schemes.In this paper we present two different applications of the cost-effective multilevel technique developed in [8] in non-standard situations. In both cases, its use has led to a drastic reduction in the computational time required for 2D numerical simulations, and hence to the ability to perform fine mesh simulation and, hence, to investigate new numerical techniques on personal computers.

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Solution of shallow water equations using fully adaptive multiscale schemes

Cited by 35 publications

References 31 publications

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction–diffusion systems

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction–diffusion systems

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

Cost-effective Multiresolution schemes for Shock Computations

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