Shi Jin

Shi Jin
University of Wisconsin–Madison | UW · Department of Mathematics

Ph.D.

About

181
Publications
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8,705
Citations
Additional affiliations
June 2009 - present
Shanghai Jiao Tong University
Position
  • Professor (Full)
September 2000 - present
University of Wisconsin–Madison
Position
  • Professor (Full)

Publications

Publications (181)
Article
The Boltzmann equation may contain uncertainties in initial/boundary data or collision kernel. To study the impact of these uncertainties, a stochastic Galerkin (sG) method was proposed in [18] and studied in the kinetic regime. When the system is close to the fluid regime (the Knudsen number is small), the method would become prohibitively expensi...
Book
Full-text available
This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growin...
Chapter
We propose a generalized polynomial chaos-based stochastic Galerkin method (gPC-sG) for the Fokker–Planck–Landau (FPL) equation with random uncertainties. The method can handle uncertainties from initial or boundary data and the neutralizing background. By a gPC expansion and the Galerkin projection, we convert the FPL equation with uncertainty int...
Article
For the Vlasov-Poisson equation with random uncertain initial data, we prove that the Landau damping solution given by the deterministic counterpart (Caglioti and Maffei, {\it J. Stat. Phys.}, 92:301-323, 1998) depends smoothly on the random variable if the time asymptotic profile does, under the smoothness and smallness assumptions similar to the...
Article
Full-text available
In this paper we study the stochastic Galerkin approximation for the linear transport equation with random inputs and diffusive scaling. We first establish uniform (in the Knudsen number) stability results in the random space for the transport equation with uncertain scattering coefficients and then prove the uniform spectral convergence (and conse...
Article
Full-text available
In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-...
Article
We introduce a sub-cell shock capturing method for scalar conservation laws built upon the Jin-Xin relaxation framework. Here, sub-cell shock capturing is achieved using the original defect measure correction technique. The proposed method exactly restores entropy shock solutions of the exact Riemann problem and, moreover, it produces monotone and...
Article
We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensio...
Article
We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus...
Article
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We study the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales. Here the uncertainty, modeled by random variables, enters the solution through initial data, while the multiple scales lead the system to its high-field or parabolic regimes. With the help of proper Lyapunov-type inequalities, under some mild conditions on the in...
Article
Full-text available
For linear transport and radiative heat transfer equations with random inputs, we develop new generalized polynomial chaos based Asymptotic-Preserving stochastic Galerkin schemes that allow efficient computation for the problems that contain both uncertainties and multiple scales. Compared with previous methods for these problems, our new method us...
Article
For the quantum kinetic system modelling the Bose-Einstein Condensate that accounts for interactions between condensate and excited atoms, we use the Chapman-Enskog expansion to derive its hydrodynamic approximations, include both Euler and Navier-Stokes approximations. The hydrodynamic approximations describe not only the macroscopic behavior of t...
Article
In this paper we consider a kinetic-fluid model for disperse two-phase flows with uncertainty. We propose a stochastic asymptotic-preserving (s-AP) scheme in the generalized polynomial chaos stochastic Galerkin (gPC-sG) framework, which allows the efficient computation of the problem in both kinetic and hydrodynamic regimes. The s-AP property is pr...
Article
The Ehrenfest dynamics, representing a quantum-classical mean-field type coupling, is a widely used approximation in quantum molecular dynamics. In this paper, we propose a time-splitting method for an Ehrenfest dynamics, in the form of a nonlinearly coupled Schr\"odinger-Liouville system. We prove that our splitting scheme is stable uniformly with...
Article
In this paper, we develop a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method for the linear semiconductor Boltzmann equation with random inputs and diffusive scalings. The random inputs are due to uncertainties in the collision kernel or initial data. We study the regularity (uniform in the Knudsen number) of the solu...
Article
Full-text available
We develop a general polynomial chaos (gPC) based stochastic Galerkin (SG) for hyperbolic equations with random and singular coefficients. Due to the singu- lar nature of the solution, the standard gPC-SG methods may suffer from a poor or even non convergence. Taking advantage of the fact that the discrete solution, by the central type finite diffe...
Article
In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-S...
Article
We introduce a new numerical strategy to solve a class of oscillatory transport PDE models which is able to captureaccurately the solutions without numerically resolving the high frequency oscillations {\em in both space and time}.Such PDE models arise in semiclassical modeling of quantum dynamics with band-crossings, and otherhighly oscillatory wa...
Article
We develop a stochastic Galerkin method for the Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) approximation in the stochastic Galerkin framework, and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related co...
Article
Full-text available
We propose a generalized polynomial chaos based stochastic Galerkin methods for scalar hyperbolic balance laws with random geometric source terms or random initial data. This method is well-balanced (WB), in the sense that it captures the stochastic steady state solution with high order accuracy. The framework of the stochastic WB schemes is presen...
Article
Full-text available
We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions - a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since...
Article
Full-text available
A Lagrangian surface hopping algorithm is implemented to study the two dimensional massless Dirac equation for Graphene with an electrostatic potential, in the semiclassical regime. In this problem, the crossing of the energy levels of the system at Dirac points requires a particular treatment in the algorithm in order to describe the quantum trans...
Article
In this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs. The schemes are asymptotic preserving (AP), in the sense that they preserve the diffusive limits of the equations in discrete setting, without requiring excessive refinement of the discretizatio...
Article
We propose a two-dimensional asymptotic-preserving scheme for linear transport equations with diffusive scalings. It is based on the time splitting developed by Jin, Pareschi and Toscani [SINUM, 2000], but takes spatial discretizations on staggered grids. Compared with the previous methods based on regular Cartesian grids, this method preserves the...
Article
We develop computational methods for high frequency solutions of general symmetric hyperbolic systems with eigenvalue degeneracies (multiple eigenvalues with constant multiplicities) in the dispersion matrices that correspond to polarized waves. Physical examples of such systems include the three-dimensional elastic waves and Maxwell equations. The...
Article
Symmetric hyperbolic systems include many physically relevant systems of PDEs such as Maxwell's equations, the elastic wave equations, and the acoustic equations [L. Ryzhik, G. Papanicolaou, and J. Keller, Wave Motion, 24 (1996), pp. 327-370]. In the current paper we extend the Gaussian beam method to efficiently compute the high frequency solution...
Article
Full-text available
We develop a class of stochastic numerical schemes for Hamilton-Jacobi equations with random inputs in initial data and/or the Hamiltonians. Since the gradient of the Hamilton- Jacobi equations gives a symmetric hyperbolic system, we utilize the generalized polynomial chaos (gPC) expansion with stochastic Galerkin procedure in random space and the...
Conference Paper
In the rarefied gas dynamics, the DSMC method is one of the most popular numerical tools. It performs satisfactorily in simulating hypersonic flows surrounding re-entry vehicles and micro-/nano- flows. However, the computational cost is expensive, especially when Kn → 0. Even for flows in the near-continuum regime, pure DSMC simulations require a n...
Article
Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation l...
Article
In this work, we propose an asymptotic-preserving Monte Carlo method for the Boltzmann equation that is more efficient than the currently available Monte Carlo methods in the fluid dynamic regime. This method is based on the successive penalty method [39], which is an improved BGK-penalization method originally proposed by Filbet and Jin [16]. Here...
Article
Full-text available
We consider a coupled system of Schr\"odinger equations, arising in quantum mechanics via the so-called time-dependent self-consistent field method. Using Wigner transformation techniques we study the corresponding classical limit dynamics in two cases. In the first case, the classical limit is only taken in one of the two equations, leading to a m...
Article
We are concerned with a coupled system describing the interaction between suspended particles and a dense fluid. The particles are modeled by a kinetic equation of Vlasov-Fokker-Planck type, and the fluid is described by the incompressible Navier-Stokes system, with variable density. The systems are coupled through drag forces. High friction regime...
Article
Full-text available
In the paper we derive a semiclassical model for surface hopping allowing quantum dynamical non-adiabatic transition between different potential energy surfaces in which cases the classical Born-Oppenheimer approximation breaks down. The model is derived using the Wigner transform and Weyl quantization, and the central idea is to evolve the entire...
Article
In the paper we derive a semiclassical model for surface hopping allowing quantum dynamical non-adiabatic transition between different potential energy surfaces in which cases the classical Born-Oppenheimer approximation breaks down. The model is derived using the Wigner transform and Weyl quantization, and the central idea is to evolve the entire...
Article
We study the quasi-random choice method (QRCM) for the Liouville equation of geometrical optics with discontinuous local wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discontinu...
Article
The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the ho-mogenization theory developed for the study of high frequency or semi-classical limit for these problems assumes no crossing of the Bloch bands, resulting in a classical Liouville equation in the limit along each...
Article
Full-text available
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 establish the relations for parameters of the Gaussian beams between the physical space (GBs) and the phase space, which motivates an efficient...
Article
We present asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime. A major challenge in this regime is that there may be no explicit expression of the local equilibrium which is the main component of classical asymptoticpreserving schemes. Inspired by [F. Filbet and S. Jin, J. Comput. Phy...
Article
An asymptotic-preserving (AP) scheme is efficient in solving multiscale problems where kinetic and hydrodynamic regimes coexist. In this article, we extend the BGK-penalization-based AP scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation (Filbet and Jin, J Comput Phys 229 (2010) 7625–7648), to its multispecies...
Article
We present a domain decomposition method on a semilinear hyperbolic system with multiple relaxation times. In the region where the relaxation time is small, an asymptotic equilibrium equation can be used for computational efficiency. An interface condition based on the sign of the characteristic speed at the interface is provided to couple the two...
Article
We propose an asymptotic-preserving (AP) scheme for kinetic equations that is efficient also in the hydrodynamic regimes. This scheme is based on the Bhantnagar-Gross-Krook (BGK) penalty method introduced by Filbet and Jin [J. Comput. Phys., 229 (2010), pp. 7625-7648], but uses the penalization successively to achieve the desired asymptotic propert...
Article
Full-text available
We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a scheme in-evitably needs an implicit discretization of the nonlinear collision operator, which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL co...
Article
Full-text available
The Dirac equation is an important model in relativistic quantum mechanics. In the semi-classical regime "≪1, even a spatially spectrally accurate time splitting method [6] requires the mesh size to be O("), which makes the direct simulation extremely expensive. In this paper, we present the Gaussian beam method for the Dirac equation. With the hel...
Article
Full-text available
Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic preserving schemes was introduced in [6] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own...
Article
Full-text available
We are interested in an Eulerian-Lagrangian model describing particulate flows. The model under study consists of the Euler system and a Vlasov-Fokker-Planck equation coupled through momentum and energy exchanges. This problem contains asymptotic regimes that make the coupling terms stiff, and lead to a limiting model of purely hydrodynamic type. W...
Article
We propose Eulerian and Lagrangian Gaussian beam methods for the Schrödinger equation with discontinuous potentials. At the quantum barriers where the potential is discontinuous, we derive suitable interface conditions to account for quantum scattering information. These scattering interface conditions are then built into the numerical fluxes in th...
Article
When using standard deterministic particle methods, point values of the computed solutions have to be recovered from their singular particle approximations by using some smoothing procedure. The choice of the smoothing procedure is rather flexible. Moreover, there is always a parameter associated with the smoothing procedure: if this parameter is t...
Article
The Vlasov-Poisson-Fokker-Planck system under the high field scaling describes the Brow-nian motion of a large system of particles in a surrounding bath where both collision and field effects (electrical or gravitational) are dominant. Numerically solving this system becomes chal-lenging due to the stiff collision term and stiff nonlinear transport...
Article
We present a class of asymptotic-preserving (AP) schemes for the nonhomogeneous Fokker–Planck–Landau (nFPL) equation. Filbet and Jin [16] designed a class of AP schemes for the classical Boltzmann equation, by penalization with the BGK operator, so they become efficient in the fluid dynamic regime. We generalize their idea to the nFPL equation, wit...
Article
Full-text available
The kinetic flux vector splitting (KFVS) scheme, when used for quantum Euler equa-tions, as was done by Yang et al [22], requires the integration of the quantum Maxwellians (Bose-Einstein and Fermi-Dirac distributions), giving a numerical flux much more compli-cated than the classical counterpart. As a result, a nonlinear 2 by 2 system that connect...
Article
We consider time-dependent (linear and nonlinear) Schrodinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approxima- tion of the solutions, or at least of the asso...
Article
Full-text available
In this paper, we propose a hybrid method coupling a Schrödinger solver and a Gaussian beam method for the numerical simulation of quantum tunneling through potential barriers or surface hopping across electronic potential energy surfaces. The idea is to use a Schrödinger solver near potantial barriers or zones where potential en-ergy surfaces cros...
Article
We consider a system coupling the incompressible Navier–Stokes equations to the Vlasov– Fokker–Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has...
Article
Full-text available
We construct a numerical scheme based on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction (GTD) to simulate the high frequency linear waves diffracted by a corner. While the reflection boundary conditions are used at the boundary, a diffraction condi-tion, based on the GTD theory, is introduced at the vert...
Article
Full-text available
In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explic...
Article
Full-text available
In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the Born-Oppenheimer approximation in the zeroth order term. The purpose of this paper is to develop the surface hopping method for the Schrodinger equation with conical crossi...
Article
Full-text available
The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffnes...
Article
In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize th...
Article
Full-text available
Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision term. A class of asymptotic preserving schemes was introduced in [6] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own...
Article
We propose a level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials. The discontinuities in the potential corresponds to potential barriers, at which incoming waves can be partially transmitted and reflected. Previously such a problem was handled by Jin and Wen using the Liouville equation – which ari...
Article
We introduce the Bloch decomposition-based time splitting spectral method to conduct numerical simulations of the (non)linear dispersive wave equations with periodic coefficients. We first consider the numerical simulations of the dynamics of nonlinear Schrödinger equations subject to periodic and confining potentials. We consider this system as a...
Article
Full-text available
As an important model in quantum semiconductor devices, the Schrodinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for hig...
Article
Full-text available
The linear Schrödinger equation with periodic potentials is an important model in solid state physics. The most efficient direct simulation using a Bloch decomposition-based time-splitting spectral method [18] requires the mesh size to be O(epsilon) where epsilon is the scaled semiclassical parameter. In this paper, we generalize the Gaussian beam...
Article
Full-text available
In this paper we extend the micro-macro decomposition based asymptotic-preserving scheme devel- oped in (3) for the single species Boltzmann equation to the multispecies problems. An asymptotic- preserving scheme for kinetic equation is very efficient in the fluid regime where the Knudsen number is small and the collision term becomes stiff. It all...
Article
Full-text available
Kinetic and hyperbolic equations contain small scales (mean free path/time, Debye length, relaxation or reaction time, etc.) that lead to various different asymptotic regimes, in which the classical numerical approximations become prohibitively expensive. Asymptotic-preserving (AP) schemes are schemes that are efficient in these asymptotic regimes....
Article
Full-text available
A novel Eulerian Gaussian beam method was developed in [8] to compute the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce an efficient semi-Eulerian implementation of this method. The new algorithm inherits the essence of the Eule-rian Gaussian beam method where the Hessian is computed through the derivativ...
Article
Full-text available
As an important model in quantum semiconductor devices, the Schrödinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for hig...
Article
In this note, we review our recent results on the Eulerian computation of high frequency waves in heterogeneous me-dia. We cover three recent methods: the moment method, the level set method, and the computational methods for in-terface problems in high frequency waves. These approaches are all based on high frequency asymptotic limits.
Article
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In this paper, we propose a uniformly second order numerical method for the discete-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 scatte...
Article
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We present a time-dependent semiclassical transport model for coher-ent pure-state scattering with quantum barriers. The model is based on a complex-valued Liouville equation, with interface conditions at quantum barriers computed from the steady-state Schrödinger equation. By retain-ing the phase information at the barrier, this coherent model ade...
Article
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We extend the Bloch-decomposition based time-splitting spectral method introduced in an earlier paper [Z. Huang, S. Jin, P. Markowich, C. Sparber, A Bloch decomposition based split-step pseudo spectral method for quantum dynamics with periodic potentials, SIAM J. Sci. Comput. 29 (2007) 515–538] to the case of (non-)linear Klein–Gordon equations. Th...
Article
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This paper reviews some recent numerical methods for hyperbolic equations with singular (discontinuous or measure-valued) coefficients. Such problems arise in wave propagation through interfaces or barriers, or nonlinear waves through singular geometries. The connection between the well-balanced schemes for shallow-water equations with discontinuou...
Article
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We study the l 1 -stability of a Hamiltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 (2005), 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the l 1 -norm under a hyperbolic CFL condition which is in consistent wit...
Article
Full-text available
We construct a numerical scheme based on the Liouville equation of geo-metric optics coupled with the Geometric Theory of Diffraction (GTD) to simulate the high frequency linear waves diffracted by a half plane. We first introduce a condition, based on the GTD theory, at the vertex of the half plane to account for the diffractions, and then build i...
Article
The solution to the Schrödinger equation is highly oscillatory when the rescaled Planck constant $\varepsilon$ is small in the semiclassical regime. A direct numerical simulation requires the mesh size to be $\emph{O}(\varepsilon)$. The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperformin...
Article
Full-text available
A linear convection equation with discontinuous coefficients arises in wave propagation through interfaces. An interface condition is needed at the interface to select a unique solution. An upwind scheme that builds this interface condition into its numerical flux is called the immersed interface upwind scheme. An l1-error estimate of such a scheme...
Article
Full-text available
We show how the level set method, developed by L.-T. Cheng, H. Liu and St. Osher [Commun. Math. Sci. 1, No. 3, 593–621 (2003; Zbl 1084.35066)], S. Jin, H. L. Liu, S. Osher and R. Tsai [J. Comput. Phys. 205, No. 1, 222–241 (2005; Zbl 1072.65132)], S. Jin and St. Osher [Commun. Math. Sci. 1, No. 3, 575–591 (2003; Zbl 1090.35116)] for the numerical co...
Article
We construct a class of numerical schemes for the Liouville equation of geometric optics coupled with the Geometric Theory of Diffractions to simulate the high frequency linear waves with a discontinuous index of refraction. In this work [S. Jin, X. Wen, A Hamiltonian-preserving scheme for the Liouville equation of geometric optics with partial tra...
Article
Full-text available
When a linear transport equation contains two scales, one diffu-sive and the other non-diffusive, it is natural to use a domain decomposition method which couples the transport equation with a diffusion equation with an interface condition. One such method was introduced by Golse, Jin and Levermore in [11], where an interface condition, which is de...
Article
Full-text available
By extending the Bloch-decomposition-based time-splitting spectral method we introduced earlier [SIAM J. Sci. Comput. 29, No. 2, 515–538 (2007; Zbl 1136.65093)], we conduct numerical simulations of the dynamics of nonlinear Schrödinger equations subject to periodic and confining potentials. We consider this system as a two-scale asymptotic problem...
Article
In this paper, we develop a numerical scheme for the interface problem in the planar symmetric radiative transfer equation with isotropic scattering. Such problems arise in the modeling of the propagation of energy density for waves in heterogeneous media with weak random fluctuation in the high frequency regime. The idea, following the earlier wor...