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2023, Vol. 43 No. 4   Published date: 28 December 2023
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• Dynamical Analysis of a Toxic-phytoplankton-zooplankton Model with Chemotaxis and Allee Effects

  Dong Yuqin, Chen Shaoyu, Dai Binxiang

  2023, 43(4): 1-28. doi: 10.3969/j.issn.1006-8074.2023.04.001

  Abstract ( 1390 )   PDF (357KB) ( 190 )   　　

  This paper demonstrates the global existence and boundedness of the solutions of a toxic- phytoplankton-zooplankton model with chemotaxis and Allee effects in a smooth bounded domian with no-flux boundary condition. This result holds for arbitrary spatial dimension and small chemotaxis coefficients. It is also proved that the positive homogeneous steady state loses its stability when the chemotactic coefficient surpasses a threshold value, and the nonhomogeneous steady states bifurcate from the homogeneous steady state. Finally a numerical simulation is performed.

• Notes on Systems of Linear Congruence Equations

  Shi Wenzhang, Liu Heguo

  2023, 43(4): 29-47. doi: 10.3969/j.issn.1006-8074.2023.04.002

  Abstract ( 668 )   PDF (206KB) ( 246 )   　　

  The Chinese Remainder Theorem is a fundamental principle in number theory. In this paper we start with the canonical form of integer matrix under modular transformations and give another proof to the Chinese Remainder Theorem. Then by using the invariant factors we give some sufficient and necessary conditions for the existence of solutions for two classes of systems of linear congruence equations with multiple variables, and further more, find the number of solutions to the systems.

• The Mordell-Weil Groups of Cubic Pencils

  Mo Jiali

  2023, 43(4): 48-58. doi: 10.3969/j.issn.1006-8074.2023.04.003

  Abstract ( 624 )   PDF (175KB) ( 169 )   　　

  In this paper we study the influences of the base points of cubic pencils on the Mordell-Weil groups. Specifically, we investigate and classify the cubic pencils with 8, 7 and 6 base points in general position, and give some applications.

• An Unconditionally Energy Stable Numerical Scheme for the Modified Phase Field Crystal Equation

  Liang Yihong, Jia Hongen

  2023, 43(4): 59-75. doi: 10.3969/j.issn.1006-8074.2023.04.004

  Abstract ( 605 )   PDF (1434KB) ( 201 )   　　

  This paper constructs a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme for the modified phase field crystal equation with periodic boundary conditions.

  The unique solvability, unconditionally energy stability and unconditionally temporal convergence of order 2 of the numerical scheme are showed by introducing a Lagrange multiplier to deal with the nonlinear terms and adopting the second-order

  Crank-Nicolson method to discrete time. Numerical experiments are given in the last section to validate the efficiency of the proposed scheme.

• Numerical Simulation Algorithms for Stochastic Differential Equations in Systems Biology

  Niu Yuanling, Chen Lin, Chen luonan

  2023, 43(4): 76-92. doi: 10.3969/j.issn.1006-8074.2023.04.005

  Abstract ( 716 )   PDF (508KB) ( 475 )   　　

  Many phenomena in systems biology, such as the biochemical reaction process, the evolution of ecosystems, the spread of infectious diseases, can be described by stochastic differential equations (SDEs). Considering the influence of randomness, stochastic differential equation models can describe the evolution of variables over time more accurately than deterministic differential equation models. However, the analytical solutions of most stochastic differential equations cannot be obtained. Even though some of them can be obtained, the forms of the solutions are usually extremely complex. One therefore requires proper numerical methods to approximate their solutions on computers. These stochastic differential equation models in systems biology usually have the properties of high dimension, high nonlinearity, and the solutions being located in a specified region. It is difficult to simulate them numerically. This paper reviews the numerical simulation algorithms of several typical models in systems biology (biochemical reaction models, ecosystem models, infectious disease models, population genetics models, cell differentiation models), and briefly introduces their advantages and disadvantages.

• A Predictor-corrector Smoothing Newton Method for Solving the Special Weighted Linear Complementarity Problem

  He Xiaorui, Tang Jingyong

  2023, 43(4): 93-105. doi: 10.3969/j.issn.1006-8074.2023.04.006

  Abstract ( 562 )   PDF (220KB) ( 159 )   　　

  In this paper, we study the method for solving the special weighted linear complementarity problem. Based on a weighted smoothing function, we reformulate the problem as a system of smooth nonlinear equations and then propose a predictor-corrector smoothing Newton method to solve it. Under some suitable conditions, we show that the algorithm has the global and local quadratic convergence properties. In particular, when the solution set is nonempty we show that the merit function sequence converges to zero. Numerical experiments demonstrate that our algorithm is effective.

•  A Semi-implicit Collocation Scheme for the Allen-Cahn Equation

  Tong Yanlei, Yao Xiaozhen, Li Mengting, Li Yiwen, Weng Zhifeng

  2023, 43(4): 106-122. doi: 10.3969/j.issn.1006-8074.2023.04.007

  Abstract ( 702 )   PDF (960KB) ( 273 )   　　

  In this paper, a semi-implicit collocation scheme and the corresponding discrete linear equation system are firstly derived for the Allen-Cahn equation by discretizing it in spatial direction with the barycentric Lagrange interpolation collocation method, in time direction with the backward Euler scheme and the Crank-Nicolson scheme respectively and treating its nonlinear term with an explicit scheme. Then, the consistency of the one-dimensional spatial semi-discrete schemes and the two-dimensional full discrete schemes are analyzed. Finally, the high accuracy and energy decreasing law of the semi-implicit collocation scheme are verified by numerical examples. Comparing with the two kinds of classical difference schemes, the numerical results show that the proposed scheme can achieve higher accuracy with fewer nodes.

• Field-Split Preconditioners for Solving Three-dimensional Reservoir Flows in Fractured Porous Media

  Yang Nian, Yang Haijian, Shao Baiqiang

  2023, 43(4): 123-140. doi: 10.3969/j.issn.1006-8074.2023.04.008

  Abstract ( 604 )   PDF (1756KB) ( 156 )   　　

  With the applications of Newton-Krylov method in solving large sparse nonlinear equations, the design of linear preconditioners plays a vital role in the whole solver. In this paper, we study the application of different combinations of field-split (FS) preconditioners based on physical and domain decomposition methods to the unsteady flow problem of fractured porous media. Under the framework of domain decomposition technology, several new FS preconditioners are considered: the additive FS preconditioner, the multiplicative FS preconditioner, the Schur-complement FS preconditioner and the constrained pressure residual (CPR) preconditioner. The corresponding subsystems are approximately solved by the restricted additive Schwarz (RAS) algorithm. In order to further improve the performance of the FS preconditioner, we designe a two-level FS preconditioner. Numerical experiments on Tianhe-2 supercomputer show that the proposed preconditioner has a good robustness and parallel scalability.