Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems

Jarczewska, K.; Glabisz, W.; Zielichowski-Haber, W.

doi:10.1016/j.acme.2014.04.008

Cited by 3 publications

(2 citation statements)

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“…Multiresolution analysis is needed to analyse the behaviour of the DG2 modes over the hierarchy of nested grids g (n) c n 164 defined on a grid element Q c . Within this scope, the multiresolution analysis procedure requires multiwavelet bases, or 165 multiwavelets, to be defined on the reference element [−1, 1] 2 = g (0) (Jarczewska et al, 2015;Kesserwani et al, 2019), from 166 a father basis,P (0) , that is actually the scaled basisP used to define the slope-decoupled DG2 expansion (Eq. (10), Section 167 2.1.1).…”

Section: Multiresolution Analysis: 2d Slope-decoupled Approaches 163mentioning

confidence: 99%

“…This paper includes a concise presentation of the numerical formulations that are accessible to engineers and modellers. Further theoretical details are given by Jarczewska et al (2015), Gerhard and Müller (2016) and Kesserwani et al (2018Kesserwani et al ( , 2019. A Fortran 2003 implementation of the solvers is also available to download from Zenodo (Sharifian and Kesserwani, 2020), and instructions for running these solvers are provided in Appendix B.…”

Section: Introductionmentioning

confidence: 99%

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(Multi)wavelets increase both accuracy and efficiency of standard Godunov-type hydrodynamic models: Robust 2D approaches

Kesserwani

Sharifian

2020

Advances in Water Resources

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Multiwavelets (MW) enable the compression, analysis and assembly of model data on a multiresolution grid withinGodunov-type solvers based on second-order discontinuous Galerkin (DG2) and first-order finite volume (FV1) methods.Multiwavelet adaptivity has been studied extensively with one-dimensional (1D) hydrodynamic models (Kesserwani et al., 2019), revealing that MWDG2 can be 20 times faster than uniform DG2 and 2 times faster than uniform FV1, while preserving the accuracy and robustness of the underlying formulation. The potential of the MWDG2 scheme has yet to be studied for two-dimensional (2D) modelling, but this requires a design that robustly and efficiently solves the 2D shallow water equations (SWE) with complex source terms and wetting and drying. This paper presents a two-dimensional MWDG2 scheme that: (1) adopts a slope-decoupled DG2 solver as a reference scheme, for its ability to deliver well-balanced piecewise-planar solutions shaped by a simplified 3-component basis; and, (2) adapts the multiresolution analysis of multiwavelets for compatibility with the slope-decoupled DG2 basis. A scaled reformulation of slope-decoupled DG2 is presented alongside two multiwavelet approaches that yield MWDG2 schemes with similar properties, and a Haar wavelet FV1 (HFV1) variant for adapting piecewise-constant model data. The performance of the adaptive HFV1 and MWDG2 solvers is explored alongside their uniform counterparts, while analysing their accuracy, efficiency, grid-coarsening ability, reliability in handling wet-dry fronts across steep bed-slopes, and ability to capture features relevant to practical hydraulic modelling. The results indicate a particular multiwavelet approach that allows the MWDG2 scheme to exploit its grid-coarsening ability for the widest range of flow types. Results also indicate that the proposed (multi)wavelet-based adaptive schemes are even more efficient for the 2D case. Accompanying model software is openly available online.

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Section: Multiresolution Analysis: 2d Slope-decoupled Approaches 163mentioning

confidence: 99%