eISSN:
 2688-1594

Electronic Research Archive

Specail issue on PDEs in fluid flow problems

Xiaoming He, Department of Mathematics and Statistics,Missouri University of Science and Technology,400 W. 12th St. Rolla, MO 65401, USA hex@mst.edu

Chunpeng Wang, School of Mathematiccs, Jilin University,Changchun 130012, China wangcp@jlu.edu.cn

Ran Zhang, School of Mathematics, Jilin University,Changchun 130012, China zhangran@jlu.edu.cn

Global weak solutions for the two-component Novikov equationSpecial Issues
Cheng He and Changzheng Qu
2020, 28(4): 1545-1562 doi: 10.3934/era.2020081 +[Abstract](15)+[HTML](14) +[PDF](411.13KB)
Abstract:

The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the \begin{document}$ H^1 $\end{document}-weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori estimates on the approximation solutions.

Viscosity robust weak Galerkin finite element methods for Stokes problemsSpecial Issues
Bin Wang and Lin Mu
2020,  doi: 10.3934/era.2020096 +[Abstract](15)+[HTML](11) +[PDF](1236.25KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

A weak Galerkin finite element method for nonlinear conservation lawsSpecial Issues
Xiu Ye Shangyou Zhang and Peng Zhu
2020,  doi: 10.3934/era.2020097 +[Abstract](15)+[HTML](11) +[PDF](413.06KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problemsSpecial Issues
ShinJa Jeong and Mi-Young Kim
2020,  doi: 10.3934/era.2020101 +[Abstract](15)+[HTML](11) +[PDF](1500.19KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

Asymptotic behavior of the one-dimensional compressible micropolar fluid modelSpecial Issues
Haibo Cui Junpei Gao and Lei Yao
2020,  doi: 10.3934/era.2020105 +[Abstract](15)+[HTML](11) +[PDF](342.13KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equationsSpecial Issues
Matthew Gardner , Adam Larios, Leo G. Rebholz, Duygu Vargun and Camille Zerfas
2020,  doi: 10.3934/era.2020113 +[Abstract](15)+[HTML](11) +[PDF](5363.96KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

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