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1. INTER-COUPLED TSUNAMI MODELLING THROUGH AN ABSORBING-GENERATING BOUNDARY

   https://doi.org/10.9753/icce.v36v.currents.38  

   Son, Sangyoung; Lynett, Patrick (December 2020, Coastal Engineering Proceedings)

   null (Ed.)

   For many practical and theoretical purposes, various types of tsunami wave models have been developed and utilized so far. Some distinction among them can be drawn based on governing equations used by the model. Shallow water equations and Boussinesq equations are probably most typical ones among others since those are computationally efficient and relatively accurate compared to 3D Navier-Stokes models. From this idea, some coupling effort between Boussinesq model and shallow water equation model have been made (e.g., Son et al. (2011)). In the present study, we couple two different types of tsunami models, i.e., nondispersive shallow water model of characteristic form(MOST ver.4) and dispersive Boussinesq model of non-characteristic form(Son and Lynett (2014)) in an attempt to improve modelling accuracy and efficiency.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/cTXybDEnfsQ 

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2. Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell’s equations in linear Lorentz media

   Jiang, Yan; Sakkaplangkul, Puttha; Bokil, Vrushali A; Cheng, Yingda; Li, Fengyan (May 2019, Journal of computational physics)

   In this paper, we consider Maxwell’s equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. This work is a continuation of our previous research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7]. The results for the numerical dispersion analysis of the reduced linear model, considered in the present paper, can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell’s equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures 

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3. A consistent adaptive level set framework for incompressible two-phase flows with high density ratios and high Reynolds numbers

   Zeng, Y; Liu, H; Gao, Q; Almgren, A; Bhalla, A; Shen, L (January 2023, Journal of computational physics)

   We develop a consistent adaptive framework in a multilevel collocated grid layout for simulating two-phase flows with adaptive mesh refinement (AMR). The conservative mo-mentum equations and the mass equation are solved in the present consistent framework. This consistent mass and momentum transport treatment greatly improves the accuracy and robustness for simulating two-phase flows with a high density ratio and high Reynolds number. The interface capturing level set method is coupled with the conservative form of the Navier–Stokes equations, and the multilevel reinitialization technique is applied for mass conservation. This adaptive framework allows us to advance all variables level by level using either the subcycling or the non-subcycling method to decouple the data ad-vancement on each level. The accuracy and robustness of the framework are validated by a variety of canonical two-phase flow problems. We demonstrate that the consistent scheme results in a numerically stable solution in flows with high density ratios(up to 106) and high Reynolds numbers(up to 106), while the inconsistent scheme exhibits non-physical fluid behaviors in these tests. Furthermore, it is shown that the subcycling and non-subcycling methods provide consistent results and that both of them can accurately resolve the interfaces of the two-phase flows with surface tension effects. Finally, a 3D breaking wave problem is simulated to show the efficiency and significant speedup of the proposed framework using AMR. 

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4. Aconsistent adaptive level set framework for incompressible two-phase flows with high density ratios and high Reynolds numbers

   Zeng, Y; Liu, H; Gao, Q; Almgren, A; Bhalla, A; Shen, L (January 2023, Journal of computational physics)

   We develop a consistent adaptive framework in a multilevel collocated grid layout for simulating two-phase flows with adaptive mesh refinement (AMR). The conservative mo-mentum equations and the mass equation are solved in the present consistent framework. This consistent mass and momentum transport treatment greatly improves the accuracy and robustness for simulating two-phase flows with a high density ratio and high Reynolds number. The interface capturing level set method is coupled with the conservative form of the Navier–Stokes equations, and the multilevel reinitialization technique is applied for mass conservation. This adaptive framework allows us to advance all variables level by level using either the subcycling or the non-subcycling method to decouple the data ad-vancement on each level. The accuracy and robustness of the framework are validated by a variety of canonical two-phase flow problems. We demonstrate that the consistent scheme results in a numerically stable solution in flows with high density ratios(up to 106) and high Reynolds numbers(up to 106), while the inconsistent scheme exhibits non-physical fluid behaviors in these tests. Furthermore, it is shown that the subcycling and non-subcycling methods provide consistent results and that both of them can accurately resolve the interfaces of the two-phase flows with surface tension effects. Finally, a 3D breaking wave problem is simulated to show the efficiency and significant speedup of the proposed framework using AMR. 

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5. Robust data-driven discovery of governing physical laws with error bars

   https://doi.org/10.1098/rspa.2018.0305  

   Zhang, Sheng; Lin, Guang (September 2018, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier–Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions. 

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