Communications on Applied Mathematics and Computation
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Singularity Formation for the General Poiseuille Flow of Nematic Liquid Crystals
Communications on Applied Mathematics and Computation - Tập 5 - Trang 1130-1147 - 2022
We consider the Poiseuille flow of nematic liquid crystals via the full
Ericksen-Leslie model. The model is described by a coupled system consisting of
a heat equation and a quasilinear wave equation. In this paper, we will
construct an example with a finite time cusp singularity due to the
quasilinearity of the wave equation, extended from an earlier result on a
special case.
Sequential Approximation of Functions in Sobolev Spaces Using Random Samples
Communications on Applied Mathematics and Computation - Tập 1 - Trang 449-466 - 2019
We present an iterative algorithm for approximating an unknown function
sequentially using random samples of the function values and gradients. This is
an extension of the recently developed sequential approximation (SA) method,
which approximates a target function using samples of function values only. The
current paper extends the development of the SA methods to the Sobolev space and
allows the...
The Hermite-Taylor Correction Function Method for Maxwell’s Equations
Communications on Applied Mathematics and Computation - - Trang 1-25 - 2023
The Hermite-Taylor method, introduced in 2005 by Goodrich et al. is highly
efficient and accurate when applied to linear hyperbolic systems on periodic
domains. Unfortunately, its widespread use has been prevented by the lack of a
systematic approach to implementing boundary conditions. In this paper we
present the Hermite-Taylor correction function method (CFM), which provides
exactly such a syst...
Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations
Communications on Applied Mathematics and Computation - - 2022
Failure-Informed Adaptive Sampling for PINNs, Part II: Combining with Re-sampling and Subset Simulation
Communications on Applied Mathematics and Computation - - Trang 1-22 - 2023
This is the second part of our series works on failure-informed adaptive
sampling for physic-informed neural networks (PINNs). In our previous work (SIAM
J. Sci. Comput. 45: A1971–A1994), we have presented an adaptive sampling
framework by using the failure probability as the posterior error indicator,
where the truncated Gaussian model has been adopted for estimating the
indicator. Here, we prese...
A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials
Communications on Applied Mathematics and Computation - Tập 1 - Trang 333-360 - 2019
In this work, we apply a semi-Lagrangian spectral method for the Vlasov–Poisson
system, previously designed for periodic Fourier discretizations, by
implementing Legendre polynomials and Hermite functions in the approximation of
the distribution function with respect to the velocity variable. We discuss
second-order accurate-in-time schemes, obtained by coupling spectral techniques
in the space–ve...
An Unconventional Divergence Preserving Finite-Volume Discretization of Lagrangian Ideal MHD
Communications on Applied Mathematics and Computation - - Trang 1-55 - 2023
We construct an unconventional divergence preserving discretization of updated
Lagrangian ideal magnetohydrodynamics (MHD) over simplicial grids. The
cell-centered finite-volume (FV) method employed to discretize the conservation
laws of volume, momentum, and total energy is rigorously the same as the one
developed to simulate hyperelasticity equations. By construction this moving
mesh method ensu...
Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation
Communications on Applied Mathematics and Computation - - Trang 1-14 - 2023
In this paper, numerical experiments are carried out to investigate the impact
of penalty parameters in the numerical traces on the resonance errors of
high-order multiscale discontinuous Galerkin (DG) methods (Dong et al. in J Sci
Comput 66: 321–345, 2016; Dong and Wang in J Comput Appl Math 380: 1–11, 2020)
for a one-dimensional stationary Schrödinger equation. Previous work showed that
penalty ...
A Multi-physics Methodology for Four States of Matter
Communications on Applied Mathematics and Computation - Tập 2 - Trang 487-514 - 2019
We propose a numerical methodology for the simultaneous numerical simulation of
four states of matter: gas, liquid, elastoplastic solids, and plasma. The
distinct, interacting physical processes are described by a combination of
compressible, inert, and reactive forms of the Euler equations, multi-phase
equations, elastoplastic equations, and resistive MHD equations. Combinations of
systems of equ...
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