arXiv subjects

Jia Zhao

Publications and source records attributed to Jia Zhao.

43 records · Page 3Linked to original sources

arXiv2017

Linear, Second order and Unconditionally Energy stable schemes for The Viscous Cahn-Hilliard Equation with hyperbolic relaxation using the invariant energy quadratization method

In this paper, we consider numerical approximations for the viscous Cahn-Hilliard equation with hyperbolic relaxation. This type of equations processes energy-dissipative structure. The main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms, allowing large time-marching step, while preserving the energy stability, i.e. the energy dissipative structure at the time-discrete level. We resolve this issue by developing two second-order time-marching schemes using the recently developed "Invariant Energy Quadratization" approach where all nonlinear terms are discretized semi-explicitly. In each time step, one only needs to solve a symmetric positive definite (SPD) linear system. All the proposed schemes are rigorously proven to be unconditionally energy stable, and the second-order convergence in time has been verified by time step refinement tests numerically. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy and efficiency of the proposed schemes.

math.NA↗

arXiv2017

Eigenvalues of Sturm-Liouville Operators with Distributional Potentials

We introduce a novel approach for dealing with eigenvalue problems of Sturm-Liouville operators generated by the differential expression \begin{equation*} Ly=\frac{1}{r}\left( -(p\left[ y^{\prime }+sy\right] )^{\prime }+sp\left[ y^{\prime }+sy\right] +qy\right) \end{equation*} which is based on norm resolvent convergence of classical Sturm-Liouville operators. This enables us to describe the continuous dependence of the $n$-th eigenvalue on the space of self-adjoint boundary conditions and the coefficients of the differential equation after giving the inequalities among the eigenvalues. Moreover, oscillation properties of the eigenfunctions are also characterized. In particular, our main results can be applied to solve a class of Sturm-Liouville problems with transmission conditions.

math.SP↗

arXiv2017

On Linear and unconditionally energy stable Algorithms for Variable Mobility Cahn-Hilliard Type Equation with Logarithmic Flory-Huggins Potential

In this paper, we consider the numerical approximations for the fourth order Cahn-Hilliard equation with concentration dependent mobility, and the logarithmic Flory-Huggins potential. One challenge in solving such a diffusive system numerically is how to develop proper temporal discretization for nonlinear terms in order to preserve the energy stability at the time-discrete level. We resolve this issue by developing a set of the first and second order time marching schemes based on a novel, called "Invariant Energy Quadratization" approach. Its novelty is that the proposed scheme is linear and symmetric positive definite because all nonlinear terms are treated semi-explicitly. We further prove all proposed schemes are unconditionally energy stable rigorously. Various 2D and 3D numerical simulations are presented to demonstrate the stability, accuracy and efficiency of the proposed schemes thereafter.

math.NA↗

arXiv2017

Numerical Approximations for a three components Cahn-Hilliard phase-field Model based on the Invariant Energy Quadratization method

How to develop efficient numerical schemes while preserving the energy stability at the discrete level is a challenging issue for the three component Cahn-Hilliard phase-field model. In this paper, we develop first and second order temporal approximation schemes based on the "Invariant Energy Quadratization" approach, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to a well-posed linear system with the symmetric positive definite operator to be solved at each time step. We rigorously prove that the proposed schemes are unconditionally energy stable. Various 2D and 3D numerical simulations are presented to demonstrate the stability and the accuracy of the schemes.

math.NA↗

arXiv2013

Packing $(2^{k+1}-1)$-order perfect binary trees into (\emph{k}+1)-connected graph

Let $G=(V,E)$ and $H$ be two graphs. Packing problem is to find in $G$ the largest number of independent subgraphs each of which is isomorphic to $H$. Let $U\subset{V}$. If the graph $G-U$ has no subgraph isomorphic to $H$, $U$ is a cover of $G$. Covering problem is to find the smallest set $U$. The vertex-disjoint tree packing was not sufficiently discussed in literature but has its applications in data encryption and in communication networks such as multi-cast routing protocol design. In this paper, we give the kind of $(k+1)$-connected graph $G'$ into which we can pack independently the subgraphs that are each isomorphic to the $(2^{k+1}-1)$-order perfect binary tree $T_k$. We prove that in $G'$ the largest number of vertex-disjoint subgraphs isomorphic to $T_k$ is equal to the smallest number of vertices that cover all subgraphs isomorphic to $T_k$. Then, we propose that $T_k$ does not have the \emph{Erdős-Pósa} property. We also prove that the $T_k$ packing problem in an arbitrary graph is NP-hard, and propose the distributed approximation algorithms.

math.CO↗

arXiv2009

A proof of the Corrected Beiter conjecture

We say that a cyclotomic polynomial Φ_{n}(x) has order three if n is the product of three distinct primes, p =11, and they proposed the Corrected Beiter conjecture: A(pqr)<=2p/3. Here we will give a proof of this conjecture.

math.NT↗

arXiv2009

On the coefficients of the cyclotomic polynomials of order three

We say that a cyclotomic polynomial Phi_{n}(x) has order three if n is the product of three distinct primes, p =11, and they proposed the Corrected Beiter conjecture: M(p)<=2p/3. Here we will give a sufficient condition for the Corrected Beiter conjecture and prove it when p=7.

math.NT↗