Fourier pseudospectral methods for 2D Boussinesq-type equations

Steinmoeller, Derek; Stastna, Marek; Lamb, Kevin G.

doi:10.1016/j.ocemod.2012.05.003

Cited by 9 publications

(7 citation statements)

References 18 publications

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“…In the present study the BR flux was avoided given its sub-optimal convergence rates (Duran and Marche, 2015). Among the other options, which can deliver optimal convergence rates (Kirby and Karniadakis, 2005;Steinmoeller et al, 2012Steinmoeller et al, , 2016, the LDG flux is chosen in this work and can be obtained by setting and (Cockburn and Shu, 1998).…”

Section: Dispersive Terms Computationmentioning

confidence: 99%

“…Following the work in Steinmoeller et al (2012), this quantity is normalized by its initial value and then recorded over time for two of the meshes (i.e. with 80 and 160 cells)…”

Section: Oscillatory Flow In a Parabolic Bowlmentioning

confidence: 99%

“…For accuracy assessment of dispersive wave behavior, a solitary wave propagating with a celerity c in the still water of depth is considered. The exact solution of the solitary wave that is similar in shape to solitons predicted by Korteweg-de Vries (KdV) equations (Steinmoeller et al, 2012), which is given by:…”

Section: Propagation Of a Solitary Wavementioning

confidence: 99%

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A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

2018

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This work extends a robust second-order Runge-Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green-Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modelling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model.

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Section: Dispersive Terms Computationmentioning

confidence: 99%

“…Following the work in Steinmoeller et al (2012), this quantity is normalized by its initial value and then recorded over time for two of the meshes (i.e. with 80 and 160 cells)…”

Section: Oscillatory Flow In a Parabolic Bowlmentioning

confidence: 99%

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A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

2018

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“…Most of the BT equations impose a smallness amplitude assumption as = O( 2 ), which is too restrictive for many applications in nearshore areas. 7,[11][12][13][14][15][16][17][18][19] However, the runtime cost of DG methods is high, given their demands for storage and evolution of local degrees of freedom within each computational cell and their restrictive CFL condition when applied with explicit Runge-Kutta (RK) time stepping. [3][4][5] The GN equations share the same characteristics of other BT models.…”

Section: Introductionmentioning

confidence: 99%

“…9-11 DG methods are becoming increasingly popular in solving BT equations. 7,[11][12][13][14][15][16][17][18][19] However, the runtime cost of DG methods is high, given their demands for storage and evolution of local degrees of freedom within each computational cell and their restrictive CFL condition when applied with explicit Runge-Kutta (RK) time stepping. These costs would even be higher when modeling wave propagation and transformation in coastal areas, where the multitude of spatial and temporal scales further increase the wave feature and complexity.Classical adaptive mesh refinement (AMR) techniques were initially used in an attempt to reduce fine resolution costs by adapting the mesh resolution.…”

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confidence: 99%

Performance study of the multiwavelet discontinuous Galerkin approach for solving the Green‐Naghdi equations

Sharifian

Hassanzadeh

Kesserwani

et al. 2019

Numerical Methods in Fluids

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Summary This paper presents a multiresolution discontinuous Galerkin (DG) scheme for the adaptive solution of Boussinesq‐type equations. The model combines multiwavelet (MW)–based grid adaptation with a DG solver based on the system of fully nonlinear and weakly dispersive Green‐Naghdi (GN) equations. The key feature of the adaptation procedure is to conduct a multiresolution analysis using MWs 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 MW coefficients controlled by a single user‐defined threshold value. The proposed adaptive multiwavelet DG solver for GN equations 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.

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On the application of 2D discrete spectral analysis in case of the KP equation

Salupere

Ratas

2018

Mechanics Research Communications

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Fourier pseudospectral methods for 2D Boussinesq-type equations

Cited by 9 publications

References 18 publications

A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

Performance study of the multiwavelet discontinuous Galerkin approach for solving the Green‐Naghdi equations

On the application of 2D discrete spectral analysis in case of the KP equation

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