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应用数学与计算数学学报
应用数学与计算数学学报

郭本瑜

季刊

1006-6330

camc@oa.shu.edu.cn

021-66137602

200444

上海市上大路99号121信箱

应用数学与计算数学学报/Journal Communication on applied mathematics and computationCSCD
查看更多>>本刊是反映应用数学与计算数学法领域中最新研究成果，促进学术交流。
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    Preface

    Qing NieChi-Wang ShuYulong XingYong-Tao Zhang...
    1-2页
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    Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

    Zachary M.MiksisYong-Tao Zhang
    3-29页
    查看更多>>摘要：Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equa-tions(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The result-ing iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local sys-tem.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for mul-tidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidi-mensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
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    Energy Stable Nodal DG Methods for Maxwell's Equations of Mixed-Order Form in Nonlinear Optical Media

    Maohui LyuVrushali A.BokilYingda ChengFengyan Li...
    30-63页
    查看更多>>摘要：In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equa-tions considered previously in Bokil et al.(J Comput Phys 350:420-452,2017)and Lyu et al.(J Sci Comput 89:1-42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell's equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlin-ear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is fur-ther applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with dE as the number of the components of the electric field,only a dE × dE nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small non-linear systems are completely decoupled over one time step,rendering very high paral-lel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accu-racy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.
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    Bifurcation Analysis Reveals Solution Structures of Phase Field Models

    Xinyue Evelyn ZhaoLong-Qing ChenWenrui HaoYanxiang Zhao...
    64-89页
    查看更多>>摘要：The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems.Here,we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models.Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or com-putationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions.To elucidate the idea,we apply this analytical approach to three representative phase field equations:the Allen-Cahn equation,the Cahn-Hilliard equation,and the Allen-Cahn-Ohta-Kawasaki system.The solution structures of these three phase field equations are also verified numerically by the homotopy continua-tion method.
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    Growth of RB Population in the Conversion Phase of Chlamydia Life Cycle

    Frederic Y.M.Wan
    90-112页
    查看更多>>摘要：Upon infecting a host cell,the reticulate body(RB)form of the Chlamydia bacteria simply proliferates by binary fission for an extended period.Available data show only RB units in the infected cells 20 hours post infection(hpi),spanning nearly half way through the development cycle.With data collected every 4 hpi,conversion to the elementary body(EB)form begins abruptly at a rapid rate sometime around 24 hpi.By modeling prolifera-tion and conversion as simple birth and death processes,it has been shown that the optimal strategy for maximizing the total(mean)EB population at host cell lysis time is a bang-bang control qualitatively replicating the observed conversion activities.However,the sim-ple birth and death model for the RB proliferation and conversion to EB deviates in a sig-nificant way from the available data on the evolution of the RB population after the onset of RB-to-EB conversion.By working with a more refined model that takes into account a small size threshold eligibility requirement for conversion noted in the available data,we succeed in removing the deficiency of the previous models on the evolution of the RB population without affecting the optimal bang-bang conversion strategy.
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    A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

    Hao LiXiangxiong Zhang
    113-141页
    查看更多>>摘要：For solving two-dimensional incompressible flow in the vorticity form by the fourth-order compact finite difference scheme and explicit strong stability preserving temporal discre-tizations,we show that the simple bound-preserving limiter in Li et al.(SIAM J Numer Anal 56:3308-3345,2018)can enforce the strict bounds of the vorticity,if the velocity field satisfies a discrete divergence free constraint.For reducing oscillations,a modified TVB limiter adapted from Cockburn and Shu(SIAM J Numer Anal 31:607-627,1994)is constructed without affecting the bound-preserving property.This bound-preserving finite difference method can be used for any passive convection equation with a divergence free velocity field.
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    Stability Analysis of Inverse Lax-Wendroff Procedure for High Order Compact Finite Difference Schemes

    Tingting LiJianfang LuPengde Wang
    142-189页
    查看更多>>摘要：This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solv-ing a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.
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    Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

    Fangyao ZhuJuntao HuangYang Yang
    190-217页
    查看更多>>摘要：In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional posi-tivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(ⅰ)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ⅱ)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time inte-grations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use Qk polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving tech-nique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.
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    Mathematical Modeling of Cell Polarity Establishment of Budding Yeast

    Yue LiuJun XieHay-Oak ParkWing-Cheong Lo...
    218-235页
    查看更多>>摘要：The budding yeast Saccharomyces cerevisiae is a powerful model system for studying the cell polarity establishment.The cell polarization process is regulated by signaling mol-ecules,which are initially distributed in the cytoplasm and then recruited to a proper location on the cell membrane in response to spatial cues or spontaneously.Polarization of these signaling molecules involves complex regulation,so the mathematical mod-els become a useful tool to investigate the mechanism behind the process.In this review,we discuss how mathematical modeling has shed light on different regulations in the cell polarization.We also propose future applications for the mathematical modeling of cell polarization and morphogenesis.
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    Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

    Weitao ChenChiu-Yen Kao
    236-256页
    查看更多>>摘要：In this paper,we review computational approaches to optimization problems of inhomoge-neous rods and plates.We consider both the optimization of eigenvalues and the localiza-tion of eigenfunctions.These problems are motivated by physical problems including the determination of the extremum of the fundamental vibration frequency and the localization of the vibration displacement.We demonstrate how an iterative rearrangement approach and a gradient descent approach with projection can successfully solve these optimization problems under different boundary conditions with different densities given.
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