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  Min Tang
Distinguished Research Fellow   Full CV
Institute of Natural Sciences and Department of Mathematics Post Address:
Shanghai Jiaotong University Room 535, Institute of Natural Sciences
Phone (Office): (86)21-54742607 Shanghai Jiaotong University
Email: tangmin@sjtu.edu.cn 800 Dongchuan road, Shanghai, 200240, China

For students:

Applications are constantly invited at graduate or postdoc level. For details, please drop me an email.

Research Interests:

Mathematical Biology; Chemotaxis, Reaction diffusion systems; traveling waves and patten formation;
Individual based models for tumor growth and its corresponding continuous model;
Multiscale Radiative Transport Equation; Anisotropic diffusion equations;
Semiclassical limit in quantum Mechanics; Singular limit problems in fluid mechanics.

List of Publications

Submitted:
  • [27]   J. G. Liu, M. Tang, L. Wang and Z. N. Zhou,
    Analysis and computation of some tumor growth models with nutrient: from the cell density models to the free boundary dynamics
    Submitted.   
    In this paper, we study the tumor growth equation along with various models for the nutrient component, including the in vitro model and the in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = n^r. For finite r, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As r\to\infty, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.
    Abstract

  • [26]   H. F. Chen, G. Y. Chen, X. Hong, H. Gao and M. Tang,
    Uniformly Convergent Scheme for RTE with Anisotropic Scattering up to the boundary and interface layers
    Submitted.   
    In this paper, we present a numerical scheme for the steady-state radiative transfer equation(RTE) with anisotropic scattering. On the one hand, for the velocity discretization, we approximate the anisotropic scattering kernel by a discrete matrix that can preserve the diffusion limit. On the other hand, for the space discretization, a uniformly convergent scheme up to the boundary or interface layer is proposed. The idea is that we first approximate the scattering coefficients as well as source by piecewise constant functions, then, in each cell, the true solution is approximated by the summation of a particular solution and a linear combinations of general solutions to the homogeneous RTE. Second-order accuracy can be observed, uniformly with respect to the mean free path up to the boundary and interface layers. The scheme works well for heterogenous medium, anisotropic sources as well as for the strong source regime.
    Abstract

  • [25]   Y. H. Wang, W. J. Ying and M. Tang,
    Uniform convergent scheme for strongly anisotropic diffusion equations with closed field lines
    Submitted.   
    In magnetized plasma, the magnetic field confines particles around field lines. The ratio between the intensity of the parallel and perpendicular viscosity or heat conduction may reach the order of 1012. When the magnetic fields have closed field lines and form a ?magnetic island?, the convergence order of most known schemes depends on the anisotropy strength. In this paper, by integration of the original differential equation along each closed field line, we introduce a simple but very efficient asymptotic preserving reformulation, which yields uniform convergence with respect to the anisotropy strength. Only slight modification to the original code is required and neither change of coordinates nor mesh adaptation is needed. Numerical examples demonstrating the performance of the new scheme are presented.
    Abstract

  • [24]   C. Emako and M. Tang,
    Well-balanced and asymptotic preserving schemes for kinetic models
    Submitted.   
    In this paper, we propose a general framework for designing nu- merical schemes that have both well-balanced (WB) and asymptotic preserving (AP) properties, for various kinds of kinetic models. We are interested in two different parameter regimes, 1) When the ratio between the mean free path and the characteristic macroscopic length tends to zero, the density can be de- scribed by (advection) diffusion type (linear or nonlinear) macroscopic models; 2) When ? = O(1), the models behave like hyperbolic equations with source terms and we are interested in their steady states. We apply the framework to three different kinetic models: neutron transport equation and its diffusion limit, the transport equation for chemotaxis and its Keller-Segel limit, and grey radiative transfer equation and its nonlinear diffusion limit. Numerical examples are given to demonstrate the properties of the schemes..
    Abstract

  • Accepted:
  • [23]   W. Sun and M. Tang,
    Macroscopic Limits of pathway-based kinetic models for E.coli chemotaxis in the exponential large gradient environment,
    Multiscale Modeling and Simulation, accepted.   
    It is of great biological interest to understand the molecular origins of chemotactic behavior of E. coli by developing population-level models based on the underlying signaling pathway dynamics. We derive macroscopic models for E.coli chemotaxis that match quantitatively with the agent-based model (SPECS) for all ranges of the spacial gradient, in particular when the chemical gradient is large such that the standard Keller-Segel model is no longer valid. These equations are derived both formally and rigorously as asymptotic limits for pathway-based kinetic equations. We also present numerical results that show good agreement between the macroscopic models and SPECS. Our work provides an answer to the question of how to determine the population-level diffusion coefficient and drift velocity from the molecular mechanisms of chemotaxis, for both shallow gradients and large gradients environments.
    Abstract

  • In Refereed Journals:
  • [22]   M. Tang, Y. H. Wang,
    Uniform convergent Tailored Finite Point method for advection-diffusion equation with discontinuous, anisotropic and vanishing diffusivity,
    Journal of Scientific Computing, Vol. 70, No.1, 272-300, January 2017.   
    We propose two Tailored Finite Point methods (TFPM) for the advection-diffusion equation with anisotropic tensor diffusivity. The diffusion coefficient can be very small in one direction in some part of the domain and be discontinuous at the interfaces. When flows advect from the vanishing-diffusivity region towards the non-vanishing diffusivity region, standard numerical schemes tend to cause spurious oscillations or negative values. Our proposed schemes have uni- form convergence in the vanishing diffusivity limit, even when the solution exhibits interface and boundary layers. When the diffusivity is along the coordinates, the positivity and maximum principle can be proved. We use the value as well as their derivatives at the grid points to construct the scheme for nonaligned case, which makes it can achieve good accuracy and convergence for the derivatives as well, even when there exhibit boundary or interface layers. Numerical experiments are presented to show the performance of the proposed scheme.
    Abstract

  • [21]   M. Tang and Y. H. Wang,
    An Asymptotic Preserving method for strongly anisotropic diffusion equations based on field line integration,
    Journal of Computational Physics, Vol. 330, No. 1, 735-748, 2017.   
    In magnetized plasma, the magnetic field confines the particles around the field lines. The anisotropy intensity in the viscosity and heat conduction may reach the order of 10 to the power 12. When the boundary conditions are periodic or Neumann, the strong diffusion leads to an ill-posed limiting problem. To remove the ill-conditionedness in the highly anisotropic diffusion equations, we introduce a simple but very efficient asymptotic preserving reformulation in this paper. The key idea is that, instead of discretizing the Neumann boundary conditions locally, we replace one of the Neumann boundary condition by the integration of the original problem along the field line, the singular terms can be replaced by O(1) terms after the integration, so that yields a well-posed problem. Small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed. Uniform convergence with respect to the anisotropy strength can be observed numerically and the condition number does not scale with the anisotropy.
    Abstract

  • [20]   B. Perthame, M. Tang, N. Vauchelet,
    Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway
    Journal of Mathematical Biology, Vol. 73, No. 5, pp. 1161-1178, 2016.   
    Kinetic-transport equation are, by now, standard models to describe the dynamics of populations of bacteria moving by run-and-tumble. Experimental observations show that bacteria increase their run duration when encountering an increasing gradient of chemotactic molecules. This led to a first class of models which heuristically include tumbling frequencies depending on the path-wise gradient of chemotactic signal. More recently, the biochemical pathways regulating the flagellar motors were uncovered. There- fore, a second class of kinetic-transport equations was derived, that takes into account an intra- cellular molecular content and which relates the tumbling frequency to this information. It turns out that the tumbling frequency depends on the chemotactic signal, and not on its gradient. For these two classes of models, macroscopic equations of Keller-Segel type, have been derived using diffusion or hyperbolic rescaling. We complete this program by showing how the first class of equations can be derived from the second class with molecular content. The main difficulty is to explain why the path-wise gradient of chemotactic signal can arise in this asymptotic process. Randomness of receptor methylation events can be included, and our approach allows to compute the tumbling frequency in presence of such a noise.
    Abstract

  • [19]   L. Tong, M. Tang and X. Yang,
    An augmented Keller-Segal model for E. coli chemotaxis in fast-varying environments,
    Communication in Mathematical Sciences, Vol.14, No.3, pp. 883-891, 2016.   
    This is a continuous study on E. coli chemotaxis under the framework of pathway-based mean- field theory (PBMFT) proposed in [G. Si, M. Tang and X. Yang, Multiscale Model. Simul., 12 (2014), 907?926], following the physical studies in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we derive an augmented Keller-Segel system with macroscopic intercellular signaling pathway dynamics. It can explain the experimental observation of phase-shift between the maxima of ligand concentration and density of E. coli in fast-varying environments at the population level. This is a necessary complement to the original PBMFT where the phase-shift can only be modeled by moment systems. Numerical simulations show the quantitative agreement of the augmented Keller-Segel model with the individual-based E. coli chemotaxis simulator.
    Abstract

  • [18]  G. Si, M. Tang and X. Yang,
    A pathway-based mean-field model for E. coli chemotaxis: mathematical derivation and keller-segel limit,
    SIAM Multiscale Modeling and Simulation, Vol. 12, No. 2, pp. 907-926, 2014.  
    A pathway-based mean-field theory (PBMFT) that incorporated the most recent quantitatively measured signaling pathway was recently proposed for the E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we formally derive a new kinetic system of PBMFT under the assumption that the methylation level is locally concentrated, whose turning operator takes into account the dynamical intracellular pathway, and hence is more physically relevant. We recover the PBMFT proposed by Si et al. as the hyperbolic limit and connect to the Keller-Segel equation as the parabolic limit of this new model. We also present the numerical evidence to show the quantitative agreement of the kinetic system with the individual based E. coli chemotaxis simulator.
    Abstract

  • [17]  B. Perthame, F. Quiros, M. Tang, N. Vauchelet,
    Derivation of a Hele-Shaw type system from a cell model with active motion,
    Interfaces and Free Boundaries, Vol. 16, pp. 489-508, 2014.  
    We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of ?mushy region? prepares the tumor invasion.
    Abstract

  • [16]  B. Perthame, M. Tang and N. Vauchelet,
    Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient,
    Mathematical Models and Methods in Applied Sciences, Vol. 24, No. 13, pp. 2601, 2014.  
    Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions. We study a simple free boundary model formed of a Hele-Shaw equation for the cell number density coupled to a diffusion equation for a nutrient. We can prove that a traveling wave solution exists with a healthy region separated from the progressing tumor by a sharp front (the free boundary) while the transition to the necrotic core is smoother. Remarkable is the pressure distribution which vanishes at the boundary of the proliferative rim with a vanishing derivative at the transition point to the necrotic core.
    Abstract

  • [15]  H. Han, M. Tang and W. Ying,
    Two uniform tailored finite point schemes for the two dimensional discrete ordinates transport equations with boundary and interface layers,
    Communication in Computational Physics, Vol. 15, No. 3, 797-826, 2014.  
    This paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime, which is valid up to the boundary and interface layers. A five-point node-centered and a four-point cell-centered tailored finite point schemes (TFPS) are introduced. The TFPS firstly approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system. Numerically, both methods can not only capture the diffusion limit, but also exhibit uniform convergence in the diffusive regime, even with boundary layers. Numerical results show that the five-point scheme has first% -order accuracy and the four-point scheme has second-order accuracy, uniformly with respect to the mean free path. Therefore the two dimensional boundary and interface layers do not need to be resolved.
    Abstract

  • [14]  D. Ying, M. Tang and S. Jin,
    The Gaussian Beam method for the wigner equation with discontinuous potentials,
    Inverse Problems and Imaging, a special issue in honor of the 60th birthday of Tony Chan. Vol. 7, No. 3, 21-,2013.  
    For the Wigner equation with discontinuous potentials, a phase space Gaussian beam (PSGB) summation method is proposed in this paper. We first derive the equations satisfied by the parameters for PSGBs and estab- lish the relations for parameters of the Gaussian beams between the physical space (GBs) and the phase space, which motivates an efficient initial data preparation thus a reduced computational cost than previous method in the literature. The method consists of three steps: 1) Decompose the initial value of the wave function into the sum of GBs and use the parameter relations to prepare the initial values of PSGBs; 2) Solve the evolution equations for each PSGB; 3) Sum all the PSGBs to construct the approximate solution of the Wigner equation. Additionally, in order to connect PSGBs at the discontinu- ous points of the potential, we provide interface conditions for a single phase space Gaussian beam. Numerical examples are given to verify the validity and accuracy of method.
    Abstract

  • [13]  M. Tang,
    A relaxation method for the pulsating front simulation of the periodic advection diffusion reaction equation.
    Communication in Mathematical Sciences, Vol. 11, No. 3,651-678, 2013.  
    In this paper, we propose a new relaxation method to study the pulsating traveling front for the one dimensional space and time periodic advection diffusion reaction equations. By introducing an additional parameter depending on time, the front position is confined around a fixed point, so that this method requires small computational domain. Moreover, the time evolution of this additional parameter gives the traveling velocity automatically and the results of the original advection diffusion reaction equation can be recovered. .
    Abstract

  • [12]  H. Han, J. Miller and M. Tang,
    A uniform convergent tailor finite point method for singularly perturbed linear ODE systems.
    Journal of Computational Mathematics. Vol.31, No.4, 422-438, 2013.  
    In scientific applications from plasma to chemical kinetics, a wide range of temporal scales can present in a system of differential equations. A major difficulty is encountered due to the stiffness of the system and it is required to develop fast numerical schemes that are able to access previously unattainable parameter regimes.
    We consider an initial-final value problem for a multi-scale singularly perturbed system of linear ordinary differential equations with discontinuous coefficients. We construct a tailored finite point method, which yields approximate solutions that converge in the maximum norm, uniformly with respect to the singular perturbation parameters, to the exact solution. A parameter-uniform error estimate in the maximum norm is also proved. The results of numerical experiments, that support the theoretical results, are reported.
    Abstract

  • [11]  W. Sun and M. Tang,
    Relaxation method for one dimensional traveling waves of singular and nonlocal equations.
    Discrete and Continuous Dynamical System - B, Vol. 18, No. 5, July 2013.  
    Recent models motivated by biological phenomena lead to non-local PDEs or systems with singularities. It has been recently understood that these systems may have traveling wave solutions that are not physically relevant. We present an original method that relies on the physical evolution to capture the "stable" traveling waves. This method allows us to obtain the traveling wave profiles and their traveling speed simultaneously. It is easy to implement, and it applies to classical differential equations as well as nonlocal equations and systems with singularities. We also show the convergence of the scheme analytically for bistable reaction diffusion equations over the whole space $\R$.
    Abstract

  • [10]  M. Tang , N. Vauchelet, I. Cheddadi, I. V. Clementel, D. Drasdo, B. Perthame,
    Composite waves for a cell population system modeling tumor growth and invasion.
    Special issue of Chinese Annals of Mathematics Ser. B delicated to Jacques-Louis Lions, Vol. 34 No.2 295--318, 2013.  
    The recent biomechanical theory of cancer growth considers solid tumors as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, the latter depending either on the local cell density (contact inhibition), on mechanical stress in the tumor, or both.
    For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling, we prove there are always traveling waves above a minimal speed and we analyse their shapes. They appear to be complex with composite shapes and discontinuities. Several small parameters allow for analytical solutions; in particular the incompressible cells limit is very singular and related to the Hele-Shaw equation. These singular traveling waves are recovered numerically.
    Abstract

  • [9]  M. Tang,
    Second order method for Isentropic Euler equation in the low Mach number regime,
    Kinetic and Related Models, Vol. 5: 1, 155-184, 2012.  
    The increasing approximation errors and severe stability requirement of the compressible Euler equations at the low Mach number regime desire schemes suitable for all Mach numbers. A second order in both space and time all speed method is developed in this paper whose idea is an improvement of the semi-implicit framework proposed in \cite{DT1}.
    The second order time discretization is based on second order Runge-Kutta method with some implicit terms. This semi-discrete framework is crucial to obtain second order as well as maintain asymptotic preserving (AP), so that can capture the right limit with coarse meshes. For the space discretization, we divide the pressure term in the momentum equation into two parts, which forms two subsystems. Different space discretizations are used for these two subsystems, one is discretized by Kurganov-Tadmor central scheme (KT), and while the other by being reformulated into an elliptic equation. The proper subsystem division depends on the time step we use and the scheme becomes explicit when the time step is small enough.
    Compared with previous semi-implicit method, this framework is simpler and natural, with only two linear elliptic equations need to be solved for each time step. It maintains the AP property of the first order method in \cite{DT1} and improves the diffusivity and accuracy significantly.

    Abstract

  • [8]  G. Nadin, B. Perthame, M. Tang,
    Can traveling waves connect two steady states? The case of nonlocal Fisher equation.
    C. R. Acad. Sci. Paris, Ser. I349, 559-557, 2011.  
    This note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in \cite{BNPR} but their asymptotic behavior was still unclear. We use here a new numerical approximation of these traveling waves which shows that some traveling waves connect the two homogeneous steady states $0$ and $1$, which is a striking fact since $0$ is dynamically unstable and $1$ is unstable in the sense of Turing.

    Abstract

  • [7]  B. Perthame, C. Schmeiser, M. Tang, N. Vauchelet,
    Traveling plateaus for a hyperbolic Keller-Segel system with logistic sensitivity; existence and branching instabilities.
    Nonlinearity, 24 1253-1270,2011. (featured article)  
    How can repulsive and attractive forces, acting on a conservative system, create stable traveling patterns or branching instabilities?
    We have proposed to study this question in the framework of the hyperbolic Keller-Segel system with logistic sensitivity. This is a model system motivated by experiments on cell communities auto-organization, a field which is also called socio-biology. We continue earlier modeling work, where we have shown numerically that branching patterns arise for this system and we have analyzed this instability by formal asymptotics for small diffusivity of the chemo-repellent.
    Here we are interested in the more general situation, where the diffusivities of both the chemo-attractant and the chemo-repellent are positive. To do so, we develop an appropriate functional analysis framework. We apply our method to two cases. Firstly we analyze steady states. Secondly we analyze traveling waves when neglecting the degradation coefficient of the chemo-repellent. This shows that in different situations the cell density takes the shape of a plateau.
    The existence of steady states and traveling plateaus are a symptom of how rich the system is and why branching instabilities can occur. Numerical tests show that large plateaus may split into smaller ones, which remain stable.

    Abstract

  • [6]  F. Cerreti, B. Perthame, C. Schmeiser, M. Tang, N. Vauchelet,
    Waves for an hyperbolic Keller-Segel model and branching instabilities.
    Mathematical Models and Methods in Applied Sciences, Vol. 21, Suppl. 825-842,2011.  
    Recent experiments for swarming of the bacteria {\em Bacillus subtilis} on nutrient rich media show that these cells are able to proliferate and spread out in colonies exhibiting complex patterns as dendritic ramifications. Is it possible to explain this process with a model that does not use local nutrient depletion?
    We present a new class of models which is compatible with the experimental observations and which predict branching instabilities and does not use nutrient limitation. These conclusions are based on numerical simulations. The most complex of these models is also the biologically most accurate but the essential effects can also be obtained in simplified versions which are amenable to analysis. An example of instability mechanism is the transition from a shock wave to a rarefaction wave in a reduced two by two hyperbolic system.
    Abstract

  • [5]  P. Degond and M. Tang,
    All speed method for the Euler equation in the low mach number limit.
    Communications in Computational Physics, 10, 1-31, 2011.  
    An all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods \cite{Jeff,HW,HWconserve},firstly,nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests aredisplayed to demonstrate its performances.
    Abstract

  • [4]  A. Decoene, A. Lorz, S. Martin, B. Maury and M. Tang,
    Simulation of self-propelled chemotactic bacteria in a stokes flow.
    ESAIM: Proceedings, 30, 105-124, 2010.  
    We prescrit a method to simulate the motion of self-propelled rigid particles in a twodimensional Stokesian fluid, taking into account chemotactic behaviour. Self-propulsion is modelled as a point force associated to each particle, placed at a certain distance from its gravity centre. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation on the whole domain, including fluid and particles: rigid motion is enforced by penalizing the strain rate tensor on the rigid domain, while incompressibility is treated by duality. This leads to a minimisation problem over unconstrained functional spaces which cari lie easily implemented from any finite element Stokes solver. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. The particles are meant to represent bacteria of the Escherichia coli type, which interact with their chemical environment through consumption of nutrients and orientation in some favorable direction. Our model takes into account the interaction with oxygen. An advection-diffusion equation on the oxygen concentration is solved in the fluid domain, with a source term accounting for oxygen consumption by the bacteria. In addition, self-propulsion is deactivated for those particles which cannot consume enough oxygen. Finally, the mode’ includes random changes in the orientation of the individual bacteria, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient and thus to reproduce chemotactic behaviour. Numerical simulations implemented with FreeFem++ are presented.
    Abstract

  • [3]  M. Tang,
    A uniform first order method for the discrete ordinate transport equation with interfaces in X,Y-geometry.
    Journal of Computational Mathematics, 27, 764-786, 2009.  
    A uniformly first order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source term by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen \cite{iterative2,iterative3}, the solution at the cell edge is approximated by its average along the edge, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies coarse meshes (meshes that do not resolve the mean free path) can be used to obtain good numerical approximations. Moreover, the uniform first order convergence with respect to the mean free path is shown numerically and the rigorous proof is discussed.
    Abstract

  • [2]  S. Jin, M. Tang and H. Han,
    A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface.
    Networks and Heterogenious Media, 4, 35-65, 2009.  
    In this paper, we propose a uniformly second order numerical method for the discrete-ordinate transport equation in the slab geometry in the diffusive regimes with interfaces. At the interfaces, the scattering coefficients have discontinuities, so suitable interface conditions are needed to define the unique solution. We first approximate the scattering coefficients by piecewise constants determined by their cell averages, and then obtain the analytic solution at each cell, using which to piece together the numerical solution with the neighboring cells by the interface conditions. We show that this method is asymptotic-preserving, which preserves the discrete diffusion limit with the correct interface condition. Moreover, we show that our method is quadratically convergent uniformly in the diffusive regime, even with the boundary layers. This is 1) the first {\it sharp} uniform convergence result for linear transport equations in the diffusive regime, a problem that involves both transport and diffusive scales; and 2) the first uniform convergence {\it valid up to the boundary} even if the boundary layers exist, so the boundary layer does not need to be resolved numerically. Numerical examples are presented to justify the uniform convergence.
    Abstract

  • [1]  P. Degond, S. Jin and M. Tang,
    On the time-splitting spectral method for the complex Ginzburg-Landau equation in the large time and space scale limit.
    SIAM J. Sci. Comp. 30, 2466-2487, 2008.  
    We are interested in the numerical approximation of the complex Ginzburg--Landau equation in the large time and space limit. There are two interesting regimes in this problem, one being the large space time limit, and one being the nonlinear Schrodinger limit. These limits have been studied analytically in, for example, \cite{TC, FH1, FH2}. We study a time splitting spectral method for this problem. In particular, we are interested in whether such a scheme is asymptotic preserving (AP) with respect to these two limits. Our results show that the scheme is AP for the first limit, but not the second one. For the large space time limit, our numerical experiments show that the scheme can capture the correct physical behavior without resolving the small scale dynamics, even for transitional problem where small and large scales coexist.
    Abstract