arXiv subjects

Xingye Yue

Publications and source records attributed to Xingye Yue.

25 records · Page 2Linked to original sources

arXiv2020

A second order accurate numerical scheme for the porous medium equation by an energetic variational approach

The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. An energetic variational approach has been studied in a recent work [6], in which the trajectory equation is obtained, and a few first order accurate numerical schemes have been developed and analyzed. In this paper, we construct and analyze a second order accurate numerical scheme in both time and space. The unique solvability, energy stability are established, based on the convexity analysis. In addition, we provide a detailed convergence analysis for the proposed numerical scheme. A careful higher order asymptotic expansion is performed and two step error estimates are undertaken. In more details, a rough estimate is needed to control the highly nonlinear term in a discrete $W^{1,\infty}$ norm, and a refined estimate is applied to derive the optimal error order. Some numerical examples are presented as well.

math.NA↗

arXiv2019

Convergence analysis of a numerical scheme for the porous medium equation by an energetic variational approach

The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. We have studied numerical methods for PME by an energetic variational approach in [C. Duan et al, J. Comput. Phys., 385 (2019) 13-32], where the trajectory equation can be obtained and two numerical schemes have been developed based on different dissipative energy laws. It is also proved that the nonlinear scheme, based on $f \log f$ as the total energy form of the dissipative law, is uniquely solvable on an admissible convex set and preserves the corresponding discrete dissipation law. Moreover, under certain smoothness assumption, we have also obtained the second order convergence in space and the first order convergence in time for the scheme. In this paper, we provide a rigorous proof of the error estimate by a careful higher order asymptotic expansion and two step error estimates. The latter technique contains a rough estimate to control the highly nonlinear term in a discrete $W^{1,\infty}$ norm, and a refined estimate is applied to derive the optimal error order.

math.NA↗

arXiv2018

Numerical methods for Porous Medium Equation by an Energetic Variational Approach

We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on $f \log f$ as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on $\frac{1}{2 f}$ as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, it is proved, by a higher order expansion technique, that both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existence methods. Due to its linear nature, the second scheme is more efficient.

math.NA↗

arXiv2018

Numerical Complete Solution for Random Genetic Drift by Energetic Variational Approach

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.

math.NA↗

arXiv2016

Behavior of different numerical schemes for population genetic drift problems

In this paper, we focus on numerical methods for the genetic drift problems, which is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time evolves. In order to find a \emph{complete solution} which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total probability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesimal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion community. Careful analysis is presented to prove that the other central scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided.

math.NA↗

arXiv2014

Multi-scale Modeling for Piezoelectric Composite Materials

In this paper, we focus on multi-scale modeling and simulation of piezoelectric composite materials. A multi-scale model for piezoelectric composite materials under the framework of Heterogeneous Multi-scale Method(HMM) is proposed. For materials with periodic microstructure, macroscopic model is derived from microscopic model of piezoelectric composite material by asymptotic expansion. Convergence analysis under the framework of homogenization theory is carried out. Moreover, error estimate between HMM solutions and homogenization solutions is derived. A 3-D numerical example of 1-3 type piezoelectric composite materials is employed to verify the error estimate.

math-ph↗

arXiv2013

Homogenization: in Mathematics or Physics?

Homogenization appeared more than 100 years ago. It is an approach to study the macro-behavior of a medium by its micro-properties. In mathematics, homogenization theory considers the limitations of the sequences of the problems and its solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size in the micro scale to that in the macro scale. So what is considered is a sequence of problems in a fixed domain while the characteristic size in micro scale tends to zero. But for the real situations in physics or engineering, the micro scale of a medium is fixed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to infinity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what means we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results.

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