查看更多>>摘要：This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(ⅰ)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for Pk(k=2,3)approximations;(ⅱ )projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(ⅲ)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh size Δx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P3 approximations.Numerical experiments are provided to verify the theoretical results.
查看更多>>摘要：In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magne-tohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimen-sional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when com-bined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.
查看更多>>摘要：In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale dis-continuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321-345,2016;Dong and Wang in J Comput Appl Math 380:1-11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numeri-cal experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.
查看更多>>摘要：In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the prob-lem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.
查看更多>>摘要：The presence of the debris in the Earth's orbit poses a significant risk to human activity in outer space.This debris population continues to grow due to ground launches,the loss of external parts from space ships,and uncontrollable collisions between objects.A compu-tationally feasible continuum model for the growth of the debris population and its spatial distribution is therefore critical.Here we propose a diffusion-collision model for the evolu-tion of the debris density in the low-Earth orbit and its dependence on the ground-launch policy.We parametrize this model and test it against data from publicly available object catalogs to examine timescales for the uncontrolled growth.Finally,we consider sensible launch policies and cleanup strategies and how they reduce the future risk of collisions with active satellites or space ships.
查看更多>>摘要：For reaction-diffusion equations in irregular domains with moving boundaries,the numeri-cal stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challeng-ing to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specific φ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.
查看更多>>摘要：In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the con-vex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretiza-tions are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.
查看更多>>摘要：als at both the upstream and downstream boundaries.In the single-species case,we prove the existence of the critical domain size and provide explicit formulas in terms of model parameters.We further derive qualitative properties of the critical domain size and show that,in some cases,the critical domain size is either strictly decreasing over all diffusion rates,or monotonically increasing after first decreasing to a minimum.We also consider competition between species differing only in their diffusion rates.For two species hav-ing large diffusion rates,we give a sufficient condition to determine whether the faster or slower diffuser wins the competition.We also briefly discuss applications of these results to competition in species whose spatial niche is affected by shifting isotherms caused by climate change.
查看更多>>摘要：Stem cell regeneration is an essential biological process in the maintenance of tissue home-ostasis;dysregulation of stem cell regeneration may result in dynamic diseases that show oscillations in cell numbers.Cell heterogeneity and plasticity are necessary for the dynamic equilibrium of tissue homeostasis;however,how these features may affect the oscillatory dynamics of the stem cell regeneration process remains poorly understood.Here,based on a mathematical model of heterogeneous stem cell regeneration that includes cell heteroge-neity and random transition of epigenetic states,we study the conditions to have oscillation solutions through bifurcation analysis and numerical simulations.Our results show vari-ous model system dynamics with changes in different parameters associated with kinetic rates,cellular heterogeneity,and plasticity.We show that introducing heterogeneity and plasticity to cells can result in oscillation dynamics,as we have seen in the homogeneous stem cell regeneration system.However,increasing the cell heterogeneity and plasticity intends to maintain tissue homeostasis under certain conditions.The current study is an initiatory investigation of how cell heterogeneity and plasticity may affect stem cell regen-eration dynamics,and many questions remain to be further studied both biologically and mathematically.
Luke AndrejekJanet BestChing-Shan ChouAman Husbands...
454-488页
查看更多>>摘要：Biology provides many examples of complex systems whose properties allow organisms to develop in a highly reproducible,or robust,manner.One such system is the growth and devel-opment of flat leaves in Arabidopsis thaliana.This mechanistically challenging process results from multiple inputs including gene interactions,cellular geometry,growth rates,and coordi-nated cell divisions.To better understand how this complex genetic and cellular information controls leaf growth,we developed a mathematical model of flat leaf production.This two-dimensional model describes the gene interactions in a vertex network of cells which grow and divide according to physical forces and genetic information.Interestingly,the model predicts the presence of an unknown additional factor required for the formation of biologically realistic gene expression domains and iterative cell division.This two-dimensional model will form the basis for future studies into robustness of adaxial-abaxial patterning.
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