#### Table of Content

15 September 2021, Volume 39 Issue 5
NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION
Qianqian Chu, Guanghui Jin, Jihong Shen, Yuanfeng Jin
2021, 39(5):  655-665.  DOI: 10.4208/jcm.2002-m2019-0213
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We consider numerical methods to solve the Allen-Cahn equation using the secondorder Crank-Nicolson scheme in time and the second-order central difference approach in space. The existence of the finite difference solution is proved with the help of Browder fixed point theorem. The difference scheme is showed to be unconditionally convergent in L norm by constructing an auxiliary Lipschitz continuous function. Based on this result, it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size. The numerical experiments also verify the reliability of the method.
A CELL-CENTERED ALE METHOD WITH HLLC-2D RIEMANN SOLVER IN 2D CYLINDRICAL GEOMETRY
Jian Ren, Zhijun Shen, Wei Yan, Guangwei Yuan
2021, 39(5):  666-692.  DOI: 10.4208/jcm.2005-m2019-0173
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This paper presents a second-order direct arbitrary Lagrangian Eulerian (ALE) method for compressible flow in two-dimensional cylindrical geometry. This algorithm has half-face fluxes and a nodal velocity solver, which can ensure the compatibility between edge fluxes and the nodal flow intrinsically. In two-dimensional cylindrical geometry, the control volume scheme and the area-weighted scheme are used respectively, which are distinguished by the discretizations for the source term in the momentum equation. The two-dimensional second-order extensions of these schemes are constructed by employing the monotone upwind scheme of conservation law (MUSCL) on unstructured meshes. Numerical results are provided to assess the robustness and accuracy of these new schemes.
A GREEDY ALGORITHM FOR SPARSE PRECISION MATRIX APPROXIMATION
Didi Lv, Xiaoqun Zhang
2021, 39(5):  693-707.  DOI: 10.4208/jcm.2005-m2019-0151
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Precision matrix estimation is an important problem in statistical data analysis. This paper proposes a sparse precision matrix estimation approach, based on CLIME estimator and an efficient algorithm GISSρ that was originally proposed for l1 sparse signal recovery in compressed sensing. The asymptotic convergence rate for sparse precision matrix estimation is analyzed with respect to the new stopping criteria of the proposed GISSρ algorithm. Finally, numerical comparison of GISSρ with other sparse recovery algorithms, such as ADMM and HTP in three settings of precision matrix estimation is provided and the numerical results show the advantages of the proposed algorithm.
A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS
Lu Zhang, Qifeng Zhang, Hai-wei Sun
2021, 39(5):  708-732.  DOI: 10.4208/jcm.2005-m2020-0029
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This paper focuses on a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations. We firstly establish a three-level finite difference scheme for the time variable followed by the linearized technique of the nonlinear term. Then the fourth-order compact finite difference method is employed to discretize the spatial variables. Hence the accuracy of the discretization is $\mathcal{O}$(τ2 + $h_1^4$ + $h_2^4$) in L2-norm, where τ is the temporal step-size, both h1 and h2 denote spatial mesh sizes in x- and y- directions, respectively. The rigorous theoretical analysis, including the uniqueness, the almost unconditional stability, and the convergence, is studied via the energy argument. Practically, the discretized system holds the block Toeplitz structure. Therefore, the coefficient Toeplitz-like matrix only requires $\mathcal{O}$(M1M2) memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O}$(M1M2(log M1 + log M2)) computational complexity by the fast Fourier transformation, where M1 and M2 denote the numbers of the spatial grids in two different directions. In order to solve the resulting Toeplitz-like system quickly, an efficient preconditioner with the Krylov subspace method is proposed to speed up the iteration rate. Numerical results are given to demonstrate the well performance of the proposed method.
MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS
Yifen Ke, Changfeng Ma
2021, 39(5):  733-754.  DOI: 10.4208/jcm.2006-m2020-0037
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In this paper, we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology. Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.
A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM
Yuping Zeng, Feng Wang, Zhifeng Weng, Hanzhang Hu
2021, 39(5):  755-776.  DOI: 10.4208/jcm.2006-m2019-0010
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In this paper, we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems. We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method, though they have essentially different bilinear forms. More precisely, we prove its reliability and efficiency for the actual error measured in the standard DG norm. We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution. Numerical results are presented to verify the theoretical analysis.
ANALYSIS ON A NUMERICAL SCHEME WITH SECOND-ORDER TIME ACCURACY FOR NONLINEAR DIFFUSION EQUATIONS
Xia Cui, Guangwei Yuan, Fei Zhao
2021, 39(5):  777-800.  DOI: 10.4208/jcm.2007-m2020-0058
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A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied. The scheme is constructed with two-layer coupled discretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN) scheme and the backward difference formula second-order (BDF2) scheme. By developing a new reasoning technique, we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers, and prove rigorously the TLCD scheme is uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiority over the CN and BDF2 schemes.