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1. DG approach to large bending plate deformations with isometry constraint

   https://doi.org/10.1142/S0218202521500044  

   Bonito, Andrea ; Nochetto, Ricardo H. ; Ntogkas, Dimitrios ( January 2021 , Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove [Formula: see text]-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments. 

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2. Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

   https://doi.org/10.1090/mcom/3842  

   Wang, Haijin ; Li, Fengyan ; Shu, Chi-Wang ; Zhang, Qiang ( November 2023 , Mathematics of Computation)

   In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the$L^2$norm of the numerical solution does not increase in time, under the time step condition$\tau \le \mathcal {F}(h/c, d/c^2)$, with the convection coefficient$c$, the diffusion coefficient$d$, and the mesh size$h$. The function$\mathcal {F}$depends on the specific IMEX temporal method, the polynomial degree$k$of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes$\tau \lesssim h/c$in the convection-dominated regime and it becomes$\tau \lesssim d/c^2$in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

    

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3. Particle‐Based Lagrangian Filtering for Locating Wave‐Generated Thermal Refugia for Coral Reefs

   https://doi.org/10.1029/2020JC016106  

   Bachman, Scott D. ; Shakespeare, Callum J. ; Kleypas, Joan ; Castruccio, Frederic S. ; Curchitser, Enrique ( July 2020 , Journal of Geophysical Research: Oceans)

   Tides and tidally generated waves play a key role in modulating the heat budget of the nearshore environment, which can significantly affect the ecology of the coastal and reef zones. The high spatial resolution required to resolve such waves in numerical models means that diagnosing wave‐generated heat fluxes (WHFs) has so far only been possible over very limited areas. Because many marine ecosystems that are affected by WHFs, such as coral reefs, are patchily distributed within domains of hundreds of kilometers, an alternative to fully wave‐resolving models is needed to identify thermal refugia and guide conservation efforts over larger regions. In this study, a one‐way nested series of regional ocean simulations has been conducted to study the role of waves in driving high‐frequency temperature variations in the Coral Triangle and to develop a method for identifying WHF in coarser‐resolution models. A filtering method using Lagrangian particles is used to separate the wave component of the flow, which is used to diagnose the wave energy and to identify locations experiencing large, high‐frequency temperature variability. These locations are shown to possess three key characteristics: large time‐mean wave kinetic energy, shallow depth, and a steep bathymetric gradient that gives access to a nearby source of cold, subthermocline water. A function of the wave kinetic energy and bathymetric slope is found to closely correlate with areas of maximal temperature variance, suggesting its use as a convenient and readily calculable metric to locate thermal refugia in observations or numerical experiments over large spatial domains.

    

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4. Analysis and application of a local discontinuous Galerkin method for the electromagnetic concentrator model

   https://doi.org/10.1002/num.23069  

   Huang, Yunqing ; Li, Jichun ; Liu, Xin ( September 2023 , Numerical Methods for Partial Differential Equations)

   In this paper, we develop a local discontinuous Galerkin (LDG) method to simulate the wave propagation in an electromagnetic concentrator. The concentrator model consists of a coupled system of four partial differential equations and one ordinary differential equation. Discrete stability and error estimate are proved for both semi‐discrete and full‐discrete LDG schemes. Numerical results are presented to justify the theoretical analysis and demonstrate the interesting wave concentration property by the electromagnetic concentrator.

    

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5. Primal Dual Methods for Wasserstein Gradient Flows

   https://doi.org/10.1007/s10208-021-09503-1  

   Carrillo, José A. ; Craig, Katy ; Wang, Li ; Wei, Chaozhen ( March 2021 , Foundations of Computational Mathematics)

   null (Ed.)

   Abstract Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method. 

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