ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 4th Dec 2022, 07:02:13pm CET

Session Overview 
Session  
MS 19a: Efficient Spectral and HighOrder Methods for HighDimensional Applications
 
Session Abstract  
In recent years, there has been a significant growth of interests in numerical methods and sci entific computing of highdimensional problems arising from different areas, including the classical gas kinetic theory, quantum mechanics, multiasset option pricing problem, uncer tainty quantification, etc. For those highdimensional applications, spectral and highorder methods are often preferred to loworder methods due to highaccuracy and lower memory request. E.g. spectral sparse grid methods and polynomial chaos are two widely used spectral methods for those high dimensional problems. Recently, deep neural network method is also found to be efficient for solving high dimensional partial differential equations and the depth of a neural network is related to the “spectral order” of the neural network approximation. This minisymposium aims to bring together active researchers in related areas to present and discuss their newest advances in both mathematical theory and numerical algorithm of efficient numerical approximations for highdimensional problems.  
Presentations  
11:50am  12:20pm
ENTROPY SATISFYING DISCONTINUOUS GALERKIN METHODS FOR NONLOCAL FOKKERPLANCK EQUATIONS Tsinghua University, China, People's Republic of We consider a class of nonlocal FokkerPlanck equations with a gradient flow structure which arises in diverse applications. The solution usually corresponds to a density distribution, hence positivity is expected. In this paper, following [H. Liu and Z. Wang. J. Comput. Phys., 328: 413–437, 2017], we design, analyze and numerically validate a high order Discontinuous Galerkin (DG) method for such problems. Both semidiscrete and fully discrete schemes admit a discrete entropy dissipation law. We construct a local flux correction upon the DDG diffusive flux, so that positivity of cell averages propagates in time. A hybrid algorithm produces nonnegative solutions for all cases under a time step constraint. 12:20pm  12:50pm
Explicit modal discontinuous Galerkin scheme for threedimensional electron Boltzmann transport equation under farfromequilibrium conditions Nanyang Technological University, Singapore The Boltzmann transport equation describes the incoherent time evolution of a quantum system consisting of a substantial number of quasiparticles (for instance electrons, phonons, excitons) and their interactions, subject to electrostatic effects in terms of the corresponding distribution function in phase space. Due to the high number of dimensions (six dimensions in phase space and one dimension in time) and their intrinsic physical properties (in particular, the nonparabolicity of the momentumenergy dispersion), the construction of an efficient numerical method represents a challenge and requires a careful balance between accuracy and computational complexity. In the current research, the development of a numerical method for solving the threedimensional electron Boltzmann transport equation (BTE) under farfromequilibrium conditions is demonstrated. An explicit modal discontinuous Galerkin (DG) scheme based on hexahedral meshes is employed for the numerical solution of the threedimensional electron BTE. A simple collisional model socalled the relaxation time approximation is used to handle the complex collisional integral operator. For the spatial discretization, the polynomial solutions are represented by scaled Legendre basis functions, and the numerical flux based on an upwind scheme is used for the solution stability. All the integrals appearing in the DG formulation are approximated with the GaussLegendre quadrature rule. A thirdorder explicit SSPRKbased temporal discretization scheme is used for the resulting semidiscrete ordinary differential equation. We study numerically the verification of the theoretical and convergence analysis. The performance of the proposed scheme is assessed by solving some benchmarks in the simulations of quantum physics. 12:50pm  1:20pm
Two Methods for Thermodynamically Consistent Model Reduction Academy of Mathematics and Systems Science, China, People's Republic of Mathematical modeling of complex dissipation systems is a challenging task. Starting from the first principle or molecular dynamics, one can formally write down some very accurate mathematical models with few assumptions on molecules, but those models are usually not computable due to huge number of freedoms or high dimensions. In this talk, we briefly introduce two approaches to build computable lowdimensional models from the underlying highdimensional model or simulated trajectory data. The thermodynamical properties are kept in both approaches. We will use the NavierStokes equations and FokkerPlanck equations for Newtonian and nonNewtonian fluids as two examples to demonstrate the methods. 
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