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Special issue on higher-order numerical methods for PDEs & applications

Yanping Chen, School of Mathematical Sciences, South China Normal University, No.55 Zhongshan Avenue West, Tianhe District, Guangzhou 510631, China yanpingchen@scnu.edu.cn

Kai Jiang, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan province, 411105, China kaijiang@xtu.edu.cn

Joaquim Rigola, Heat and Mass Transfer Technological Center (CTTC)Universitat Politècnica de Catalunya - Barcelona Tech (UPC)ESEIAAT, C/ Colom, 11 - 08222 Terrassa (Barcelona), Spain quim@cttc.upc.edu

Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensionsSpecial Issues
Haolei Wang and Lei Zhang
2020, 28(1): 405-421 doi: 10.3934/era.2020023 +[Abstract](11)+[HTML](9) +[PDF](1697.67KB)

Granular materials are heterogenous grains in contact, which are ubiquitous in many scientific and engineering applications such as chemical engineering, fluid mechanics, geomechanics, pharmaceutics, and so on. Granular materials pose a great challenge to predictability, due to the presence of critical phenomena and large fluctuation of local forces. In this paper, we consider the quasi-static simulation of the dense granular media, and investigate the performances of typical minimization algorithms such as conjugate gradient methods and quasi-Newton methods. Furthermore, we develop preconditioning techniques to enhance the performance. Those methods are validated with numerical experiments for typical physically interested scenarios such as the jamming transition, the scaling law behavior close to the jamming state, and shear deformation of over jammed states.

A hybridized weak Galerkin finite element scheme for general second-order elliptic problemsSpecial Issues
Guanrong Li, Yanping Chen and Yunqing Huang
2020, 28(2): 821-836 doi: 10.3934/era.2020042 +[Abstract](11)+[HTML](5) +[PDF](377.34KB)

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equationsSpecial Issues
Changling Xu and Tianliang Hou
2020, 28(2): 897-910 doi: 10.3934/era.2020047 +[Abstract](19)+[HTML](8) +[PDF](310.06KB)

In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the \begin{document}$ H^1 $\end{document}-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshesSpecial Issues
Jianguo Huang and Sen Lin
2020, 28(2): 911-933 doi: 10.3934/era.2020048 +[Abstract](17)+[HTML](11) +[PDF](1531.97KB)

This paper is concerned with a \begin{document}$ C^0P_2 $\end{document} time-stepping virtual element method (VEM) for solving linear wave equations on polygonal meshes. The spatial discretization is carried out by the VEM while the temporal discretization is obtained based on the \begin{document}$ C^0P_2 $\end{document} time-stepping approach, leading to a fully discrete method. The error estimates in the \begin{document}$ H^1 $\end{document} semi-norm and \begin{document}$ L^2 $\end{document} norm are derived after detailed derivation. Finally, the numerical performance and efficiency of the proposed method is illustrated by several numerical experiments.

Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbationSpecial Issues
Shan Jiang, Li Liang, Meiling Sun and Fang Su
2020, 28(2): 935-949 doi: 10.3934/era.2020049 +[Abstract](17)+[HTML](10) +[PDF](434.86KB)

A multiscale finite element method is proposed for addressing a singularly perturbed convection-diffusion model. In reference to the a priori estimate of the boundary layer location, we provide a graded recursion for the mesh adaption. On this mesh, the multiscale basis functions are able to capture the microscopic boundary layers effectively. Then with a global reduction, the multiscale functional space may efficiently reflect the macroscopic essence. The error estimate for the adaptive multiscale strategy is presented, and a high-order convergence is proved. Numerical results validate the robustness of this novel multiscale simulation for singular perturbation with small parameters, as a consequence, high accuracy and uniform superconvergence are obtained through computational reductions.

High-order energy stable schemes of incommensurate phase-field crystal modelSpecial Issues
Kai Jiang and Wei Si
2020, 28(2): 1077-1093 doi: 10.3934/era.2020059 +[Abstract](27)+[HTML](19) +[PDF](3164.56KB)

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the \begin{document}$ L^2 $\end{document} gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.

The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutionsSpecial Issues
Yin Yang, Sujuan Kang and Vasiliy I. Vasil'ev
2020, 28(3): 1161-1189 doi: 10.3934/era.2020064 +[Abstract](23)+[HTML](12) +[PDF](436.96KB)

In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the \begin{document}$ L^{\infty} $\end{document}- and \begin{document}$ L^{2}_{\omega^{\alpha, \beta}} $\end{document}-norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.

Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy modelSpecial Issues
Jiangshan Wang, Lingxiong Meng and Hongen Jia
2020, 28(3): 1191-1205 doi: 10.3934/era.2020065 +[Abstract](9)+[HTML](8) +[PDF](361.28KB)

In this paper, we construct a modular grad-div stabilization method for the Navier-Stokes/Darcy model, which is based on the first order Backward Euler scheme. This method does not enlarge the accuracy of numerical solution, but also can improve mass conservation and relax the influence of parameters. Herein, we give stability analysis and error estimations. Finally, by some numerical experiment, the scheme our proposed is shown to be valid.

A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid modelSpecial Issues
Zhonghua Qiao and Xuguang Yang
2020, 28(3): 1207-1225 doi: 10.3934/era.2020066 +[Abstract](17)+[HTML](7) +[PDF](1610.81KB)

In this work, a multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is proposed to solve a coupled chemotaxis-fluid model. In the evolution equation of the proposed LBM, Beam-Warming (B-W) scheme is used to enhance the numerical stability. In numerical experiments, at first, the comparison between the classical LBM and the present LBM with B-W scheme is carried out by simulating blow up phenomenon of the Keller-Segel (K-S) model. Numerical results show that the stability of the present LBM with B-W scheme is better than the classical one. Then, the second order convergence rate of the proposed B-W scheme is verified in the numerical study of the coupled Navier-Stokes (N-S) K-S model. Finally, through solving the coupled chemotaxis-fluid model, the formation of falling bacterial plumes is numerically investigated. Numerical results agree well with existing ones in the literature.

A robust adaptive grid method for singularly perturbed Burger-Huxley equationsSpecial Issues
Li-Bin Liu, Ying Liang, Jian Zhang and Xiaobing Bao
2020, 28(4): 1439-1457 doi: 10.3934/era.2020076 +[Abstract](15)+[HTML](16) +[PDF](466.79KB)

In this paper, an adaptive grid method is proposed to solve one-dimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arc-length function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.

Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problemsSpecial Issues
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li and Shang Liu
2020, 28(4): 1459-1486 doi: 10.3934/era.2020077 +[Abstract](16)+[HTML](12) +[PDF](2201.39KB)

This paper aims at investigating the convergence and quasi- optimality of an adaptive finite element method for control constrained nonlinear elliptic optimal control problems. We derive a posteriori error estimation for both the control, the state and adjoint state variables under controlling by \begin{document}$ L^2 $\end{document}-norms where bubble function is a wonderful tool to deal with the global lower error bound. Then a contraction is proved before the convergence is proposed. Furthermore, we find that if keeping the grids sufficiently mildly graded, we can prove the optimal convergence and the quasi-optimality for the adaptive finite element method. In addition, some numerical results are presented to verify our theoretical analysis.

A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstructionSpecial Issues
Zexuan Liu, Zhiyuan Sun and Jerry Zhijian Yang
2020, 28(4): 1487-1501 doi: 10.3934/era.2020078 +[Abstract](16)+[HTML](11) +[PDF](5508.22KB)

We numerically investigate the superconvergence property of the discontinuous Galerkin method by patch reconstruction. The convergence rate \begin{document}$ 2m+1 $\end{document} can be observed at the grid points and barycenters in one dimensional case with uniform partitions. The convergence rate \begin{document}$ m + 2 $\end{document} is achieved at the center of the element faces in two and three dimensions. The meshes are uniformly partitioned into triangles/tetrahedrons or squares/hexahedrons. We also demonstrate the details of the implementation of the proposed method. The numerical results for all three dimensional cases are presented to illustrate the propositions.

Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr mediumSpecial Issues
Xuefei He, Kun Wang and Liwei Xu
2020, 28(4): 1503-1528 doi: 10.3934/era.2020079 +[Abstract](15)+[HTML](11) +[PDF](1371.93KB)

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.

Error estimates for second-order SAV finite element method to phase field crystal modelSpecial Issues
Liupeng Wang and Yunqing Huang
2021, 29(1): 1735-1752 doi: 10.3934/era.2020089 +[Abstract](906)+[HTML](347) +[PDF](883.1KB)

In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order \begin{document}$ O(\tau^2+h^2) $\end{document} in the sense of \begin{document}$ L^2 $\end{document}-norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equationsSpecial Issues
Guoliang Zhang, Shaoqin Zheng and Tao Xiong
2021, 29(1): 1819-1839 doi: 10.3934/era.2020093 +[Abstract](912)+[HTML](440) +[PDF](522.1KB)

In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [21], with a high order RKEI method [7], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.

Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element methodSpecial Issues
Ying Liu, Yanping Chen, Yunqing Huang and Yang Wang
2021, 29(1): 1859-1880 doi: 10.3934/era.2020095 +[Abstract](704)+[HTML](289) +[PDF](2296.84KB)

The mathematical model of a semiconductor device is described by a coupled system of three quasilinear partial differential equations. The mixed finite element method is presented for the approximation of the electrostatic potential equation, and the characteristics finite element method is used for the concentration equations. First, we estimate the mixed finite element and the characteristics finite element method solution in the sense of the \begin{document}$ L^q $\end{document} norm. To linearize the full discrete scheme of the problem, we present an efficient two-grid method based on the idea of Newton iteration. The two-grid algorithm is to solve the nonlinear coupled equations on the coarse grid and then solve the linear equations on the fine grid. Moreover, we obtain the \begin{document}$ L^{q} $\end{document} error estimates for this algorithm. It is shown that a mesh size satisfies \begin{document}$ H = O(h^{1/2}) $\end{document} and the two-grid method still achieves asymptotically optimal approximations. Finally, the numerical experiment is given to illustrate the theoretical results.

A conforming discontinuous Galerkin finite element method on rectangular partitionsSpecial Issues
Yue Feng, Yujie Liu, Ruishu Wang and Shangyou Zhang
2021, 29(3): 2375-2389 doi: 10.3934/era.2020120 +[Abstract](568)+[HTML](265) +[PDF](316.24KB)

This article presents a conforming discontinuous Galerkin (conforming DG) scheme for second order elliptic equations on rectangular partitions. The new method is based on DG finite element space and uses a weak gradient arising from local Raviart Thomas space for gradient approximations. By using the weak gradient and enforcing inter-element continuity strongly, the scheme maintains the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this new conforming DG scheme is significantly reduced compared to other existing DG methods. Error estimates of optimal order are established for the corresponding conforming DG approximations in various discrete Sobolev norms. Numerical results are presented to confirm the developed convergence theory.

2020 Impact Factor: 1.833
5 Year Impact Factor: 1.833



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