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Session Overview
MS 19a: Efficient Spectral and High-Order Methods for High-Dimensional Applications
Wednesday, 14/July/2021:
11:50am - 1:50pm

Session Chair: Xiaoliang Wan
Session Chair: Haijun Yu
Virtual location: Zoom 5

Session Abstract

In recent years, there has been a significant growth of interests in numerical methods and sci-

entific computing of high-dimensional problems arising from different areas, including the

classical gas kinetic theory, quantum mechanics, multi-asset option pricing problem, uncer-

tainty quantification, etc. For those high-dimensional applications, spectral and high-order

methods are often preferred to low-order methods due to high-accuracy 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 high-dimensional problems.

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11:50am - 12:20pm


Hui Yu

Tsinghua University, China, People's Republic of

We consider a class of nonlocal Fokker-Planck 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 semi-discrete 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 non-negative solutions for all cases under a time step constraint.

12:20pm - 12:50pm

Explicit modal discontinuous Galerkin scheme for three-dimensional electron Boltzmann transport equation under far-from-equilibrium conditions

Satyvir Singh, Marco Battiato

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 non-parabolicity of the momentum-energy 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 three-dimensional electron Boltzmann transport equation (BTE) under far-from-equilibrium conditions is demonstrated. An explicit modal discontinuous Galerkin (DG) scheme based on hexahedral meshes is employed for the numerical solution of the three-dimensional electron BTE. A simple collisional model so-called 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 Gauss-Legendre quadrature rule. A third-order explicit SSP-RK-based temporal discretization scheme is used for the resulting semi-discrete 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

Haijun Yu

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 low-dimensional

models from the underlying high-dimensional model or simulated trajectory data. The

thermodynamical properties are kept in both approaches. We will use the Navier-Stokes equations

and Fokker-Planck equations for Newtonian and non-Newtonian fluids as two examples to

demonstrate the methods.

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