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Archive : Numerical Analysis and PDE

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Dr. Chi-Wang Shu, Brown UniversityDr. Chi-Wang Shu, Brown UniversityEWG336Title: Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs<br><br> Description: When designing high order schemes for solving time-dependent PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous. It is then important to have a high order time discretization to main the stability properties of the semi-discrete schemes. In this talk we discuss several classes of high order time discretization, including the implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion equation) implicitly and the less stiff term (e.g. the convection term in such an equation) explicitly, for which strong stability can be proved under the condition that the time step is upper-bounded by a constant under suitable conditions, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes. Numerical examples will be given to demonstrate the performance of these schemes. 11/14/2019 8:30:00 PM11/14/2019 9:30:00 PMFalse
Dr. Shangyou Zhang, University of DelawareDr. Shangyou Zhang, University of DelawareEWG336Title:Conforming discontinuous Galerkin finite element methods for 2nd and 4th order elliptic equations<br><br> Abstract: In the weak formulation of the discontinuous Galerkin finite element method for 2nd order elliptic equations, three inter-element face-integral terms are added to the continuous weak formulation, penalizing the discontinuity of the function and the normal derivative along the inter-element boundary. Based on the principle of weak Galerkin, we design a conforming discontinuous Galerkin method eliminating all three face-integral terms of the traditional discontinuous Galerkin method so that the weak formulation is identical to that of the continuous finite element. Similarly the conforming discontinuous Galerkin method eliminates all six face-integral terms of the discontinuous Galerkin method in solving 4th order elliptic equations. We show the conforming discontinuous Galerkin method converges at the optimal order. 11/7/2019 8:30:00 PM11/7/2019 9:30:00 PMFalse
Dr. Weiwei Hu, University of GeorgeDr. Weiwei Hu, University of GeorgeEWG336Title: TBA<br><br> Description: TBA10/24/2019 7:30:00 PM10/24/2019 8:30:00 PMFalse
Professor Giovanni Russo, University of Catania Professor Giovanni Russo, University of CataniaEWG336Title: Multi-scale numerical modeling of sorption kinetics <br><br> Abstract:<br>The trapping of diffusing particles by either a single or a distribution of moving traps is an interesting topic that has been employed to model a variety of different real problems in chemistry, physics and biology. Here we study the dynamics of diffusing particles in a domain with an oscillating bubble. Laboratory experiments provide evidence of a non monotone behavior in time of the concentration of particles by a detector located behind the bubble, under suitable experimental condition. A comprehensive explanation of the phenomenon is not yet fully available.<br><br> The particles are attracted and trapped near the surface of the bubble. The basic mathematical model is a drift-diffusion model, where the particles diffuse and feel the potential of the bubble when they are near its surface. A tentative explanation of the mechanism is based on two-carrier dynamics. <br><br> The numerical simulation of the system presents two multi-scale challenges. One is spatial: the range of the bubble potential is confined within a few microns at the bubble surface, while the bubble radius is of the order of a millimeter, so a fully resolved solution would be too expensive. The second challenge is on the time scale: the bubble oscillates with a frequency of the order of 100 Hz, while the diffusion time scale is of the order of 1000 seconds, this requiring at least one million time steps to fully resolve the problem in time.<br><br> A reduced model is derived to solve the multi-scale problem it space for the single carrier dynamics: the interaction with the bubble is modeled as a very thin layer, with a particle surface density proportional to the local density in the bulk, near the bubble. In the rest of the domain the particle density satisfies just a diffusion equation, with suitable boundary conditions on the bubble, deduced from conservation properties. The model is carefully tested on problems in 1D, 2D planar and 3D axis-symmetric geometry. The equation is discretized on a regular Cartesian mesh, using a ghost-point approach, and solved by Crank-Nicolson scheme. The implicit step is efficiently solved by a suitably adapted multi-grid method.<br><br> The amplitude of the bubble oscillations is small compared to the bubble radius. We take advantage of this fact by replacing the time dependent position by a suitable time dependent velocity at the bubble surface. Because of the low Reynolds number, the velocity distribution is computed by Stokes approximation.<br><br> The multi-scale challenge in time, as well as the multi-scale model for multi-carrier dynamics are still under investigation.10/17/2019 7:30:00 PM10/17/2019 8:30:00 PMFalse

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