We construct new HLL-type moving-water equilibria preserving upwind schemes for the one-dimensional Saint-Venant system of shallow water
equations with nonflat bottom topography. The designed first- and second-order schemes are tested on a number of numerical examples, in which we verify the well-balanced property as well as the ability of the proposed schemes
to accurately capture small perturbations of moving-water steady states.
Of concern is the scenario of a heat equation on a domain that contains a thin layer, on which the thermal conductivity is drastically different
from that in the bulk. The multi-scales in the spatial variable and the thermal conductivity lead to computational difficulties, so we may think of the
thin layer as a thickless surface, on which we impose "effective boundary conditions" (EBCs). These boundary conditions not only ease the computational
burden, but also reveal the effect of the inclusion. In this paper, by considering
the asymptotic behavior of the heat equation with interior inclusion subject to
Dirichlet boundary condition, as the thickness of the thin layer shrinks, we derive, on a closed curve inside a two-dimensional domain, EBCs which include
a Poisson equation on the curve, and a non-local one. It turns out that the
EBCs depend on the magnitude of the thermal conductivity in the thin layer,
compared to the reciprocal of its thickness.
Vibrations of a beam can be described as an Euler-Bernoulli beam,
or as a Rayleigh beam or as a Timoshenko beam. In this paper, we establish
the existence of periodic solutions in time for a damped Rayleigh beam model
with time delay, which is treated as a bifurcation parameter. The main proof
is based on a Lyapunov-Schmidt reduction together with the classical implicit
function theorem. Moreover, we give a sufficient condition for a direction of
While the spread of COVID-19 in China is under control, the pandemic is developing rapidly around the world. Due to the normal migration
of population, China is facing the high risk from imported cases. The potential
specific medicine and vaccine are still in the process of clinical trials. Currently,
controlling the impact of imported cases is the key to prevent new outbreak
of COVID-19 in China. In this paper, we propose two impulsive systems to
describe the impact of multilateral imported cases of COVID-19. Based on the
published data, we simulate and analyze the epidemic trends under different
control strategies. In particular, we compare four different scenarios and show
the corresponding medical burden. The results can be useful in designing appropriate control strategy for imported cases in practice.
We investigate the M-eigenvalues of the Riemann curvature tensor
in the higher dimensional conformally flat manifold. The expressions of M-eigenvalues and M-eigenvectors are presented in this paper. As a special case,
M-eigenvalues of conformal flat Einstein manifold have also been discussed,
and the conformal the invariance of M-eigentriple has been found. We also
reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely. We also give an example to compute the M-eigentriple of de Sitter spacetime which is well-known in general relativity.
Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional
case. Unlike classical numerical methods, such as finite difference method and
finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty
method. In the work, we conduct a comparison study for elliptic problems with
four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin
method and deep Ritz method. In the former, the PDE residual is minimized
in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin
method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear
dependence of dimensionality even for smooth solutions and deep Galerkin
method can also outperform deep Ritz method for low-regularity solutions.
Besides, in some cases, when the boundary condition can be implemented in
an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.
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