A hybrid spectral/DG method for solving the phase-averaged ocean wave equation: Algorithm and validation

Yildirim, Battalgazi; Karniadakis, George Em

doi:10.1016/j.jcp.2012.04.013

Cited by 7 publications

(7 citation statements)

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“…The overall scheme 103 has exponential convergence rather than algebraic convergence 104 typical of low-order schemes. We have verified the exponential 105 convergence in both the geophysical and spectral spaces in previ-106 ous work (Yildirim and Karniadakis, 2012). The low-order methods 107 associated with strong numerical dissipation and phase errors 108 smear out the amplitude of solution and shift the position of it.…”

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confidence: 69%

“…Karniadakis / Ocean Modelling xxx (2014) Yildirim and Karniadakis (2012) using the HISWA experiment. 961 We used a coarse triangular mesh in which different Jacobi polyno-962 mial orders ðp ¼ 1; 3; 5Þ are expanded over the triangular elements.…”

Section: Q1mentioning

confidence: 99%

“…86 On the other hand, the finite element (FE) and finite volume (FV) 87 methods that work on a general grid offer an accurate and efficient 88 algorithm. Recent works have incorporated these methods into the 89 wave models to handle complex coastal boundaries (Hsu et al,90 2005; Qi et al, 2009 Recently we introduced a new numerical method for the spec-93 tral ocean wave equations (Yildirim and Karniadakis, 2012), which 94 is distinctively different from previous approaches (Booij et al, 95 1999;Hsu et al, 2005;Qi et al, 2009;Zijlema, 2010) and employs 96 high-order discretization. Specifically, we compute the spectral 97 space derivatives by Fourier-collocation while we discretize the 98 physical space using a discontinuous Galerkin (DG) method 99 (Yildirim and Karniadakis, 2012;Karniadakis and Sherwin, 1999; 100 Hesthaven and Warburton, 2007;Cockburn et al, 2000).…”

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confidence: 99%

“…We refer the interested readers to 1146 Young's book (Young, 1999) for further discussion and references. for the derivative term along the frequency axis, using absorbing 1172 boundary layer approach , was proposed in our pre-1173 vious work (Yildirim and Karniadakis, 2012), and it will be 1174 described in Appendix B.3. Let us discretize the wave direction h 1175 and the transformed frequency direction ðfÞ as The differentiation matrix for an even num-1190 ber of collocation points (Lagrangian polynomial if q ¼ 0) reads in a 1191 dummy coordinate z (either h or f to obtain the above flux 1192 derivatives) The atan mapping (Boyd, 1987(Boyd, , 2001 can be defined in ½Àp; p 1210 for the spectral direction (Yildirim and Karniadakis, 2012;Yildirim, 2012).…”

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confidence: 99%

“…for the derivative term along the frequency axis, using absorbing 1172 boundary layer approach , was proposed in our pre-1173 vious work (Yildirim and Karniadakis, 2012), and it will be 1174 described in Appendix B.3. Let us discretize the wave direction h 1175 and the transformed frequency direction ðfÞ as The differentiation matrix for an even num-1190 ber of collocation points (Lagrangian polynomial if q ¼ 0) reads in a 1191 dummy coordinate z (either h or f to obtain the above flux 1192 derivatives) The atan mapping (Boyd, 1987(Boyd, , 2001 can be defined in ½Àp; p 1210 for the spectral direction (Yildirim and Karniadakis, 2012;Yildirim, 2012). 1221 We have lection of polynomials of degree up to k in the element X such that Karniadakis and Sherwin (1999) 1288 the complete polynomial space for a triangular region can be con-1306 structed fðpqÞj0 6 p; q; p 6 P1; p þ q 6 P 2 ; P 1 6 P 2 g. The orthogonal 1307 expansion functions in terms of Jacobi polynomials are defined in 1308 Karniadakis and Sherwin (1999): The discretized equations will be now reduced to an ordinary 1321 differential equation (ODE) for the expansion coefficients u pq 1322 M pq;pq du pq ðtÞ dt With regards to temporal discretization, we employed second-1347 and third-order (Gottlieb and Shu, 1998), as well as the fourth-1348 order explicit (5-stages) (Spiteri and Steven, 2001) Strong Stability…”

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confidence: 99%

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Stochastic simulations of ocean waves: An uncertainty quantification study

Yildirim

Karniadakis

2015

Ocean Modelling

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19Keywords: 20 Phase-averaged model 21Ocean wave modeling 22Uncertainty quantification 23Generalized polynomial chaos 24Sparse grid collocation 25Sensitivity analysis 26Karhunen-Loeve decomposition Q2 27 2 8 a b s t r a c t 29 The primary objective of this study is to introduce a stochastic framework based on generalized polyno-30 mial chaos (gPC) for uncertainty quantification in numerical ocean wave simulations. The techniques we 31 present can be easily extended to other numerical ocean simulation applications. We perform stochastic 32 simulations using a relatively new numerical method to simulate the HISWA (Hindcasting Shallow Water 33 Waves) laboratory experiment for directional near-shore wave propagation and induced currents in a 34 shallow-water wave basin. We solve the phased-averaged equation with hybrid discretization based 35 on discontinuous Galerkin projections, spectral elements, and Fourier expansions. We first validate the 36 deterministic solver by comparing our simulation results against the HISWA experimental data as well 37 as against the numerical model SWAN (Simulating Waves Nearshore). We then perform sensitivity anal-38 ysis to assess the effects of the parametrized source terms, current field, and boundary conditions. We 39 employ an efficient sparse-grid stochastic collocation method that can treat many uncertain parameters 40 simultaneously. We find that the depth-induced wave-breaking coefficient is the most important param-41 eter compared to other tunable parameters in the source terms. The current field is modeled as random 42 process with large variation but it does not seem to have a significant effect. Uncertainty in the source 43 terms does not influence significantly the region before the submerged breaker whereas uncertainty in 44 the incoming boundary conditions does. Considering simultaneously the uncertainties from the source 45 terms and boundary conditions, we obtain numerical error bars that contain almost all experimental 46 data, hence identifying the proper range of parameters in the action balance equation.47

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confidence: 69%

Section: Q1mentioning

confidence: 99%