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February 2003 - present

February 2002 - January 2003

## Publications

Publications (131)

We make a brief historical review of the moment model reduction for the kinetic equations, particularly Grad’s moment method for Boltzmann equation. We focus on the hyperbolicity of the reduced model, which is essential for the existence of its classical solution as a Cauchy problem. The theory of the framework we developed in the past years is the...

We propose a discontinuous least squares finite element method for solving the Helmholtz equation. The method is based on the L2 norm least squares functional with the weak imposition of the continuity across the interior faces as well as the boundary conditions. We minimize the functional over the discontinuous polynomial spaces to seek numerical...

To close the moment model deduced from kinetic equations, the canonical approach is to provide an approximation to the flux function not able to be depicted by the moments in the reduced model. In this paper, we propose a brand new closure approach with remarkable advantages than the canonical approach. Instead of approximating the flux function, t...

We propose an Hermite spectral method for the Fokker-Planck-Landau (FPL) equation. Both the distribution functions and the collision terms are approximated by series expansions of the Hermite functions. To handle the complexity of the quadratic FPL collision operator, a reduced collision model is built by adopting the quadratic collision operator f...

We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with weak imposition of continuity across interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical solutions. T...

We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with the weak imposition of the continuity across the interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical...

We derive a nonlinear moment model for radiative transfer equation in 3D space, using the method to derive the nonlinear moment model for the radiative transfer equation in slab geometry. The resulted 3D HMPN model enjoys a list of mathematical advantages, including global hyperbolicity, rotational invariance, physical wave speeds, spectral accurac...

We propose a discontinuous least squares finite element method for solving the linear elasticity. The approximation space is obtained by patch reconstruction with only one unknown per element. We apply the $L^2$ norm least squares principle with the weak imposition of the continuity across the interior faces to the stress–displacement formulation....

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize...

We make a brief historical review to the moment model reduction to the kinetic equations, particularly the Grad's moment method for Boltzmann equation. The focus is on the hyperbolicity of the reduced model, which is essential to the existence of its classical solution as a Cauchy problem. The theory of the framework we developed in last years is t...

We propose a numerical method to solve the Monge-Ampere equation which admits a classical convex solution. The Monge-Ampere equation is reformulated into an equivalent first-order system. We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials. This space allows us to solve the first ord...

Linear models for the radiative transfer equation have been well developed, while nonlinear models are seldom investigated even for slab geometry due to some essential difficulties. We have proposed a moment model in MPN for slab geometry which combines the ideas of the classical PN and MN model. Though the model is far from perfect, it was demonst...

We study the approximation of the radiative transfer equation with a relatively few moments in the spherically symmetric case. We propose a three-moment model based on choosing the beta distribution as the ansatz for the specific intensity. This ansatz enables our model to capture the anisotropy in the distribution function. The characteristic stru...

This paper is concerned with the approximation of the radiative transfer equation for a grey medium in the slab geometry by the moment method. We develop a novel moment model inspired by the classical PN model and MN model. The new model takes the ansatz of the M1 model as the weight function and follows the primary idea of the PN model to approxim...

A discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence‐free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as a...

We model the Knudsen layer in Kramers' problem by linearized high order hyperbolic moment system. Due to the hyperbolicity, the boundary conditions of the moment system is properly reduced from the kinetic boundary condition. For Kramers' problem, we give the analytical solutions of moment systems. With the order increasing of the moment model, the...

We propose an arbitrary-order discontinuous Galerkin method for second-order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error est...

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation to the gradient in a piecewisely irrotational polynomial space. Then together with the numerical gradient, we...

A Godunov-type finite volume scheme on unstructured triangular grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate. We show the direct observation that the model is a not Galilean invariant system. At the cell boundary, the modified Harten-Lax-van Leer (HLL) approximate Riemann solver is adopted to calculate...

In this paper, we develop a patch reconstruction finite element method for the Stokes problem. The weak formulation of the interior penalty discontinuous Galerkin is employed. The proposed method has a great flexibility in velocity–pressure space pairs whose stability properties are confirmed by the inf–sup tests. Numerical examples show the applic...

In this paper, we investigate the effect of the filter for the hyperbolic moment equations(HME) [15] of the Vlasov-Poisson equations and propose a novel quasi time- consistent filter to suppress the numerical recurrence effect. By taking properties of HME into consideration, the filter preserves a lot of physical properties of HME, including Galile...

We propose a discontinuous least squares finite element method for solving the linear elasticity. The approximation space is obtained by patch reconstruction with only one unknown per element. We apply the L 2 norm least squares principle to the stress-displacement formulation based on discontinuous approximation with normal continuity across the i...

We study the approximation of the radiative transfer equation with a relatively few moments in the spherically symmetric case. We propose a three-moment model based on choosing the beta distribution as the ansatz for the specific intensity. This ansatz enables our model to capture the anisotropy in the distribution function. The characteristic stru...

We propose a robust approximate solver for the hydro-elastoplastic solid material, a general constitutive law extensively applied in explosion and high speed impact dynamics, and provide a natural transformation between the fluid and solid in the case of phase transitions. The hydrostatic components of the solid is described by a family of general...

We propose a new least squares finite element method to solve the Poisson equation. By using a piecewisely irrotational space to approximate the flux, we split the classical method into two sequential steps. The first step gives the approximation of flux in the new approximation space and the second step can use flexible approaches to give the pres...

We adopt an interior penalty discontinuous Galerkin method using a patch reconstructed approximate space to solve the elliptic eigenvalue problems, including both the second and fourth order problem in 2D and 3D. It is a direct extension of the method recently proposed to solve corresponding boundary value problems, that the optimal error estimates...

This paper is concerned with the approximation of the radiative transfer equation for a grey medium in the slab geometry by the moment method. We develop a novel moment model inspired by the classical $P_N$ model and $M_N$ model. The new model takes the ansatz of the $M_1$ model as the weight function and follows the primary idea of the $P_N$ model...

We propose a discontinuous Galerkin(DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both L2 norm and DG energy norm are obtained, without the typical constra...

A discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence-free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as a...

We propose a Hermite-Galerkin spectral method to numerically solve the spatially homogeneous Fokker-Planck-Landau equation with singular quadratic collision model. To compute the collision model, we adopt a novel approximation formulated by a combination of a simple linear term and a quadratic term very expensive to evaluate. Using the Hermite expa...

We propose an approximate solver for multi-medium Riemann problems with materials described by a family of general Mie–Grüneisen equations of state, which are widely used in practical applications. The solver provides the interface pressure and normal velocity by an iterative method. The well-posedness and convergence of the solver is verified with...

In this paper, we develop a patch reconstruction finite element method for the Stokes problem. The weak formulation of the interior penalty discontinuous Galerkin is employed. The proposed method has a great flexibility in velocity-pressure space pairs whose stability properties are confirmed by the inf-sup tests. Numerical examples show the applic...

We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and se...

A numerical method on unstructured Eulerian grids was proposed recently by the authors for two-phase compressible flows with ideal gases [Adv. Appl. Math. Mech., 8(2), 2016, pp. 187--212]. We extend the method therein to the case with fluids described by the Mie-Gr\"uneisen equation of state, which is widely used in practical applications, and prop...

We proposed a piecewise quadratic reconstruction method, which is in an integrated style, for finite volume schemes to scalar conservation laws. This quadratic reconstruction is parameter-free, is of third order accuracy for smooth functions, and is flexible on structured and unstructured grids. The finite volume schemes with the new reconstruction...

We point out that the quantum Grad's 13-moment system [R. Yano, Physica A: Statistical Mechanics and its Applications, 416:231-241, 2014] is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, we follow the new theory developed in [Z....

We extend to three-dimensional space the approximate M_2 model for the slab geometry studied in our previous paper. The B_2 model therein, as a special case of the second order extended quadrature method of moments (EQMOM), is proved to be globally hyperbolic. The model we proposed here extends EQMOM to multiple dimensions following the idea to app...

The moment method is not only a modeling tool that gives macroscopic fluid equations by reducing kinetic equations, but also a numerical method for solving kinetic equations. It has been the subject of rapid development, and in recent years has acquired widespread applications. In this paper, we review and summarize the research development of mome...

In numerical approaches for the Boltzmann equation, the discrete velocity model and the moment method are formally very different. In this paper, we try to show the intrinsic connection between these two approaches. Precisely, the Grad type moment method with appropriate closure can be regarded as a discrete velocity model with some adaptivities in...

We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition in (Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even p...

Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems, and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collis...

Linear reconstruction based on local cell-averaged values is the most commonly adopted technique to achieve a second-order accuracy when one uses the finite volume scheme on unstructured grids. For solutions with discontinuities appearing in such as conservation laws, a certain limiter has to be applied to the predicted gradient to prevent numerica...

Extended hydrodynamic models for carrier transport are derived from the semiconductor Boltzmann equation with relaxation time approximation of the scattering term, by using the globally hyperbolic moment method and the moment-dependent relaxation time. Incorporating the microscopic relaxation time and the applied voltage bias, a formula is proposed...

We study the acceleration of steady-state computation for microflow, which is modeled by the high-order moment models derived recently from the steady-state Boltzmann equation with BGK-type collision term. By using the lower-order model correction, a novel nonlinear multi-level moment solver is developed. Numerical examples verify that the resultin...

We develop a numerical method to simulate a two-phase compressible flow with sharp phase interface on Eulerian grids. The scheme makes use of a levelset to depict the phase interface numerically. The overall scheme is basically a finite volume scheme. By approximately solving a two-phase Riemann problem on the phase interface, the normal phase inte...

We consider the simplest member of the hierarchy of the extended quadrature method of moments (EQMOM), which gives equations for the zeroth-, first-, and second-order moments of the energy density of photons in the radiative transfer equations in slab geometry. First we show that the equations are well-defined for all moment vectors consistent with...

We first investigate the structure of the systems derived from the gPC based stochastic Galerkin method for the nonlinear hyperbolic systems with random inputs. This method adopts a generalized Polynomial Chaos (gPC) approximations in the stochastic Galerkin framework, but such approximations to the nonlinear hyperbolic systems do not necessarily y...

By a further study of the mechanism of the hyperbolic regularization of the
moment system for Boltzmann equation proposed in [Z. Cai, Y. Fan, R. Li, Comm.
Math. Sci. 11(2): 547-571, 2013], we point out that the key point is treating
the time and space derivative in the same way. Based on this understanding, a
uniform framework to derive globally hy...

The extended magnetohydrodynamic models are derived based on the moment closure of the Vlasov-Maxwell (VM) equations. We adopt the Grad type moment expansion which was firstly proposed for the Boltzmann equation. A new regularization method for the Grad’s moment system was recently proposed to achieve the globally hyperbolicity so that the local we...

In a recent paper [Z.-N. Cai, Y.-W. Fan, and R. Li. Tech Report, Institude of
Math, Peking Univeristy(2013)], it was revealed that a modified 13-moment
system taking intrinsic heat fluxes as variables, instead of the heat fluxes
along the coordinate vectors which is adopted in the classical Grad 13-moment
system, attains some additional advantages...

A multi-scale method for the hyperbolic systems governing sediment transport
in subcritical case is developed. The scale separation of this problem is due
to the fact that the sediment transport is much slower than flow velocity. We
first derive a zeroth order homogenized model, and then propose a first order
correction. It is revealed that the fir...

By a further investigation on the structure of the coefficient matrix of the
globally hyperbolic regularized moment equations for Boltzmann equation in [Z.
Cai, Y. Fan and R. Li, Comm. Math. Sci., 11 (2013), pp. 547-571], we propose a
uniform framework to carry out model reduction to general kinetic equations, to
achieve certain moment system. With...

Aimed to simulate the propagation of blast wave with high density ratio and high pressure ratio produced by strong explosion in the air, a two dimensional numerical program is written in which the problem is treated as a two-medium compressible flow with sharp material interface in Eulerian grids. In this method, the finite volume method is used to...

We propose an approximate second order maximum entropy ($M_2$) model for
radiative transfer in slab geometry. The model is based on the ansatz of the
specific intensity in the form of a $\Beta$-distribution. This gives us an
explicit form in its closure. The closure is very close to that of the maximum
entropy, thus an approximation of the $M_2$ mo...

We investigate the discretization of of an electron–optical phonon scattering using a finite volume method. The discretization is conservative in mass and is essentially based on an energy point of view. This results in a discrete scattering system with elegant mathematical features, which are fully clarified. Precisely the discrete scattering matr...

We point out that the thermodynamic equilibrium is not an interior point of
the hyperbolicity region of Grad's 13-moment system. With a compact expansion
of the phase density, which is compacter than Grad's expansion, we derived a
modified 13-moment system. The new 13-moment system admits the thermodynamic
equilibrium as an interior point of its hy...

Making use of the Whittaker-Shannon interpolation formula with shifted
sampling points, we propose in this paper a well-posed semi-discretization of
the stationary Wigner equation with inflow BCs. The convergence of the
solutions of the discrete problem to the continuous problem is then analysed,
providing certain regularity of the solution of the...

By the moment closure of the Boltzmann transport equation, the extended hydrodynamic models for electron transport have been derived in Cai et al. (J Math Phys 53:103503, 2012). With the numerical scheme developed in Li et al. (2012) recently, it has been demonstrated that the derived extended hydrodynamic models can capture the major features of t...

We develop a nonlinear multigrid method to solve the steady state of
microflow, which is modeled by the high order moment system derived recently
for the steady-state Boltzmann equation with ES-BGK collision term. The solver
adopts a symmetric Gauss-Seidel iterative scheme nested by a local Newton
iteration on grid cell level as its smoother. Numer...

We develop the dimension-reduced hyperbolic moment method for the Boltzmann equation, to improve solution efficiency using a numerical regularized moment method for problems with low-dimensional macroscopic variables and high-dimensional microscopic variables. In the present work, we deduce the globally hyperbolic moment equations for the dimension...

Based on the well-posedness of the stationary Wigner equation with inflow
boundary conditions given in (A. Arnold, H et al. J. Math. Phys., 41, 2000), we
prove without any additional prerequisite conditions that the solution of the
Wigner equation with symmetric potential and inflow boundary conditions will be
symmetric. This improve the result in...

The numerical method used to solve hyperbolic conservation laws is often an explicit scheme. As a commonly used technique to improve the quality of numerical simulation, the
$h$
-adaptive mesh method is adopted to resolve sharp structures in the solution. Since the computational costs of altering the mesh and solving the PDEs are comparable, too...

A globally hyperbolic high-order moment method of the Boltzmann transport equation (BTE) is proposed in [1], [2], and here it is extended for the BTE with the electron-phonon scattering term to simulate a silicon nano-wire (SNW). Convergence with respect to the order of the moment system and the characteristics of SNW including the I-V curve are st...

In this paper, we propose a globally hyperbolic regularization to the general
Grad's moment system in multi-dimensional spaces. Systems with moments up to an
arbitrary order are studied. The characteristic speeds of the regularized
moment system can be analytically given and only depend on the macroscopic
velocity and the temperature. The structure...

The hyperbolic moment system is derived for the Boltzmann equation with the ES-BGK collision term and wall boundary conditions. The wall boundary conditions we proposed for the moment system have the same number of constraints as required based on the characteristic structures of the hyperbolic moment systems. A numerical scheme is then developed t...

In this paper, we present a regularization to 1D Grad's moment system to
achieve global hyperbolicity. The regularization is based on the observation
that the characteristic polynomial of the Jacobian of the flux in Grad's moment
system is independent of the intermediate moments. The method is not relied on
the form of the collision at all, thus th...

We propose a robust finite volume scheme to numerically solve the shallow water equations on complex rough topography. The major difficulty of this problem is introduced by the stiff friction force term and the wet/dry interface tracking. An analytical integration method is presented for the friction force term to remove the stiffness. In the vicin...

In this paper, we extend the method in Cai et al. (J Math Phys 53:103503, 2012) to derive a class of quantum hydrodynamic models for the density-functional theory (DFT). The most popular implement of DFT is the Kohn–Sham equation, which transforms a many-particle interacting system into a fictitious non-interacting one-particle system. The Kohn–Sha...

A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in [6]. For numerically solving the high order hyperbolic moment system therein, we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in [8], to validate the moment system of the Wigner equation...

In this paper, we derive the quantum hydrodynamics models based on the moment
closure of the Wigner equation. The moment expansion adopted is of the Grad
type firstly proposed in \cite{Grad}. The Grad's moment method was originally
developed for the Boltzmann equation. In \cite{Fan_new}, a regularization
method for the Grad's moment system of the B...

In this paper, we propose a moment method to numerically solve the Vlasov
equations using the framework of the NRxx method developed in [6, 8, 7] for the
Boltzmann equation. Due to the same convection term of the Boltzmann equation
and the Vlasov equation, it is very convenient to use the moment expansion in
the NRxx method to approximate the distr...

In this paper, we propose a numerical regularized moment method to solve the
Boltzmann equation with ES-BGK collision term to simulate polyatomic gas flows.
This method is an extension to the polyatomic case of the method proposed in
[9], which is abbreviated as the NRxx method in [8]. Based on the form of the
Maxwellian, the Laguerre polynomials o...

We propose an efficient heterogeneous multiscale finite element method based on a local least-squares reconstruction of the effective matrix using the data retrieved from the solution of cell problems posed on the vertices of the triangulation. The method achieves high order accuracy for high order macroscopic solver with essentially the same cost...

We introduce two residual type a posteriori error estimators for second-order elliptic partial differential equations with its right-hand side in L
p
(1 < p ⩽ 2) space. Both estimators are proved to yield global upper and local lower bounds for the W
1,p
seminorm of the error. We adopt the estimators as the indicators in h-mesh adaptive method to s...

A recent work of Li et al. [Numer. Math. Theor. Meth. Appl., 1(2008), pp. 92-112] proposed a finite volume solver to solve 2D steady Euler equations. Although the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the shock region, the overshoot or undershoot phenomenon can still be observed. More-over, the numerical ac...