14 February 2024, Volume 46 Issue 1

    

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  AN EXTENDED GOLDEN RATIO PROXIMAL ALTERNATING DIRECTION METHOD OF MULTIPLIERS FOR SEPARABLE CONVEX OPTIMIZATION

  Yan Xihong, Li Hao, Wang Chuanlong, Chen Hongmei, Yang Junfeng

  Mathematica Numerica Sinica. 2024, 46(1): 1-16. https://doi.org/10.12286/jssx.j2023-1056

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  Alternating direction method of multipliers(ADMM) is one of the classical algorithms for solving separable convex optimization problems, but it cannot guarantee the convergence of primal iterates and its subproblems can be computationally demanding. In order to ensure convergence and improve computational efficiency, the golden ratio proximal ADMM using convex combination technique is proposed, where the convex combination factor $\psi$ is the key parameter. Based on the golden ratio proximal ADMM, we enlarge parameter $\psi$ and propose an extended golden ratio proximal ADMM(EgrpADMM). Under very mild assumptions, we establish the global convergence and $\mathcal{O}(1/N)$ ergodic sublinear convergence rate in terms of function value residual and constraint violation of EgrpADMM. Furthermore, the algorithm can achieve $\mathcal{O}(1/N^2)$ ergodic convergence when any of the separable subfunctions of the objective function is strongly convex. Finally, we demonstrate the performance of the proposed algorithms via numerical experiments.
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  ON THE EQUIVALENCE OF PROJECTED ITERATIVE METHODS AND MODULUS-BASED MATRIX SPLITTING ITERATIVE METHODS FOR HORIZONTAL LINEAR COMPLEMENTARITY PROBLEMS

  Cao Yang, Yang Gengchen, Shen Qinqin, Zhou Chencan

  Mathematica Numerica Sinica. 2024, 46(1): 17-37. https://doi.org/10.12286/jssx.j2022-1012

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  Horizontal linear complementarity problem (HLCP) is one of the important generalization of the famous linear complementarity problem (LCP). The projected iterative method and the modulus-based matrix splitting iterative method are two recent proposed very effective methods for solving the HLCP. The research in this paper shows that although the deriving principles of these two methods are different, they are equivalent under certain conditions. In particular, when the parameter matrix Ω in the modulus-based matrix splitting iteration methods is taken as a specific positive diagonal matrix, the projected Jacobi method, the projected Gauss-Seidel method and the projected SOR method are equivalent to the modulus-based Jacobi iteration method, the accelerated modulus-based Gauss-Seidel iteration method and the accelerated modulus-based SOR iteration method, respectively. In addition, for the general positive diagonal matrix Ω, the equivalence of these two methods is also studied. Finally, a numerical example is presented to verify the obtained theoretical results.
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  POSITIVITY-PRESERVING SCHEME FOR DIFFUSION EQUATION WITH SECOND ORDER TIME ACCURACY ON DEFORMDE MESHES

  Xie Chenyuan, Lan Bin, Yang Dexian, Li Haiyan

  Mathematica Numerica Sinica. 2024, 46(1): 38-46. https://doi.org/10.12286/jssx.j2022-1041

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  A two-layer nonlinear finite volume scheme for 2D unstationary diffusion equations is constructed on deformed meshes, which based on a two-point nonlinear discrete scheme of continuous diffusion flux. The scheme uses the idea of Crank-Nicolson (C-N) method to achieve second-order accuracy for time evolution. Since the transpose of the resulting algebraic system of coefficient matrix is an M-matrix, it is guaranteed that the scheme preserves positivity. The existence of discrete solution for the present scheme is proved by using Brouwer fixed-point theorem. Numerical results illustrate that the scheme has secondorder accuracy with a larger time step.
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  NUMERICAL STUDY ON A SCATTERING of MULTIPLE OBSTACLES BELOW SEA SURFACE

  Wang Jue, Qi Yan

  Mathematica Numerica Sinica. 2024, 46(1): 47-78. https://doi.org/10.12286/jssx.j2022-1043

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  In this paper, the scattering problem of multiple obstacles under the sea surface in twodimensional space is studied theoretically and numerically. By analyzing the characteristics of the scattering problem, using the Helmholtz equation, and combining different boundary conditions and radiation conditions, the mathematical model is established, and the uniqueness of the scattering problem is proved. Based on the potential theory and the indirect integral equation method, the integral representation of the fields in different regions and the integral boundary equation of the density function on the boundary is obtained. By introducing potential operator, the integral domain is truncated, and the operator equation on the bounded domain is obtained. For the established boundary integral equation system, the numerical scheme is constructed using the Nyström method, and the convergence of the numerical solution is proved. Finally, numerical experiments are used to verify the correctness and effectiveness of the theory. Furthermore, numerical experiments are designed to analyze the effects of different parameters on the scattering problem.
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  A DIVERGENCE-FREE STABLE QUADRATIC FINITE VOLUME METHOD SCHEME FOR THE STOKES EQUATIONS

  Zhang Jiehua, Han Minghua

  Mathematica Numerica Sinica. 2024, 46(1): 79-98. https://doi.org/10.12286/jssx.j2023-1047

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  A Lagrange quadratic finite volume method scheme for solving the Stokes equation is constructed on triangular meshes in this paper. The piecewise continuous quadratic finite element space and the discontinuous linear finite element space is used as the trial space for velocity and pressure of the Stokes equation respectively, so that the discrete velocity solution of the finite volume method satisfies the local mass conservation on the macro-element triangular element, and the finite element space pair is naturally satisfied with the so-called inf-sup condition. By adopting the special dual partition and the special mapping, the finite volume method scheme for solving the Stokes equation is transformed into the corresponding finite element method. The unconditional stability (or inf-sup condition) of the finite volume method scheme (without the geometric constraints of the triangular meshes) and the optimal-order error estimates in the $\mathbf{H}^1$-norm for velocity are obtained. Finally, numerical experiments show the validity of the theoretical results and the effectiveness of the finite volume method in the numerical simulation of computational fluid dynamics.
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  UNCONDITIONAL SUPERCONVERGENCE ANALYSIS OF ECONOMICAL FINITE DIFFERENCE STREAMLINED DIFFUSION METHOD FOR NONLINEAR CONVECTION-DOMINANTED DIFFUSION EQUATION

  Shi Dongyang, Zhang Lingen

  Mathematica Numerica Sinica. 2024, 46(1): 99-115. https://doi.org/10.12286/jssx.j2023-1048

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  In this article, the backward Euler (BE) fully discrete finite element method of the economical finite difference streamlined diffusion (EFDSD) method for nonlinear convection-dominated diffusion equation is mainly investigated and the superconvergence of order $O(h^2+\tau)$ in $H^1$ norm is derived without the restriction between the time step $\tau$ and the mesh size $h$. Firstly, a time discrete system is established to split the error into two parts, which are the temporal error and spatial error, and with the help of mathematical induction, the regularity of the time discrete system is reduced by the temporal error. Then the finite element solution in $W^{0, \infty}$ norm is bounded by the spatial error and the unconditional superclose and global superconvergence results are gained in $H^1$ norm through interpolation post-processing technique. Lastly, a numerical example is provided to verify the correctness of the theoretical analysis and the effectiveness of the method.
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  CONVERGENCE THEOREM OF THE GLOBAL ERROR OF THE FAST MULTIPOLE METHOD FOR POTENTIAL PROBLEMS IN 3-D

  Liu Zhizhao, Meng Wenhui

  Mathematica Numerica Sinica. 2024, 46(1): 116-128. https://doi.org/10.12286/jssx.j2023-1092

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  The fast multipole method (FMM) can accelerate the iterative solver of the large dense linear equations arising from many physical problems. This article is concerned with the convergence of the FMM for three dimensional potential problems. Firstly, derive the expression of the global error, and then give a novel estimate of the error bound. Secondly, the result is applied to the adaptive octree structure, and the specific convergence order is obtained. Finally, an illustrative example is provided to test the proposed results. The method of this paper can also be used to estimate the error of the FMM for elastostatic problems and Stokes flow problems.