High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

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 problem,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.

Investigation of Acoustomagnetoelectric Effect in Bandgap Graphene by the Boltzmann Transport Equation

We study the acoustomagnetoelectric (AME) effect in two-dimensional graphene with an energy bandgap using the semiclassical Boltzmann transport equation within the hypersound regime, (where represents the acoustic wavenumber and is the mean free path of the electron). The Boltzmann transport equation and other relevant equations were solved analytically to obtain an expression for the AME current density, consisting of longitudinal and Hall components. Our numerical results indicate that both components of the AME current densities display oscillatory behaviour. Furthermore, geometric resonances and Weiss oscillations were each defined using the relationship between the current density and Surface Acoustic Wave (SAW) frequency and the inverse of the applied magnetic field, respectively. Our results show that the AME current density of bandgap graphene, which can be controlled to suit a particular electronic device application, is smaller than that of (gapless) graphene and is therefore, more suited for nanophotonic device applications.

Analysis of anomalous transport with temporal fractional transport equations in a bounded domain

Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations.The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α(α ∈(0, 1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result,if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.

TRANSPORTATION COST-INFORMATION INEQUALITY FOR A STOCHASTIC HEAT EQUATION DRIVEN BY FRACTIONAL-COLORED NOISE

In this paper,we prove Talagrand’s T2 transportation cost-information inequality for the law of stochastic heat equation driven by Gaussian noise,which is fractional for a time variable with the Hurst index H∈(1/2,1),and is correlated for the spatial variable.The Girsanov theorem for fractional-colored Gaussian noise plays an important role in the proof.

SPIRAL SOLUTION TO THE TWO-DIMENSIONAL TRANSPORT EQUATIONS

The existence of spiral solution for the two-dimensional transport equations is considered in the present paper. Based on the notion of generalized solutions in the sense of Lebesgue-stieltjes integral, the global weak solution of transport equations which includes δ-shocks and vacuum is constructed for some special initial data.

WEAK SOLUTIONS OF MONGE-AMPRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION

This paper concerns the weak solutions of some Monge-Amp^re type equa- tions in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Ampere type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.

NUMERICAL SIMULATION OF SINGLE-GROUP, STEADY STATE AND ISOTROPIC NEUTRON TRANSPORT EQUATION IN DIFFUSIVE REGIMES

We present an algorithm for numerical solution of transport equation in diffusive regimes, in which the transport equation is nearly singular and its solution becomes a solution of a diffusion equation. This algorithm, which is based on the Least-squares FEM in combination with a scaling transformation, presents a good approximation of a diffusion operator in diffusive regimes and guarantees an accurate discrete solution. The numerical experiments in 2D and 3D case are given, and the numerical results show that this algorithm is correct and efficient.

Spherical harmonics method for neutron transport equation based on unstructured-meshes

Based on a new second-order neutron transport equation, self-adjoint angular flux (SAAF) equation, the spherical harmonics (PN) method for neutron transport equation on unstructured-meshes is derived. The spherical harmonics function is used to expand the angular flux. A set of differential equations about the spatial variable, which are coupled with each other, can be obtained. They are solved iteratively by using the finite element method on un- structured-meshes. A two-dimension transport calculation program is coded according to the model. The numerical results of some benchmark problems demonstrate that this method can give high precision results and avoid the ray effect very well.

A MODIFIED CRANK－NICOLSON SCHEME FOR THE INITIAL－BOUNDARY VALUE PROBLEM OF SUPERTHERMAL ELECTRON TRANSPORT EQUATION

The superthermal electron transport equation is a degenerate andnon-local evolutionary equation. In this paper, a modified Crank-Nicolsonscheme is constructed for the numerical solution. It is proved that the schemeis uniquely solvable and unconditionally

Nanofluids Transport Model Based on Fokker-Planck Equation and the Convection Heat Transfer Calculation

In current research about nanofluid convection heat transfer, random motion of nanoparticles in the liquid distribution problem mostly was not considered. In order to study on the distribution of nanoparticles in liquid, nanofluid transport model in pipe is established by using the continuity equation, momentum equation and Fokker-Planck equation. The velocity distribution and the nanoparticles distribution in liquid are obtained by numerical calculation, and the effect of particle size and particle volume fraction on convection heat transfer coefficient of nanofluids is analyzed. The result shows that in high volume fraction （ 0 _-- 0.8% ）, the velocity distribution of nanofluids characterizes as a ＂cork-shaped＂ structure, which is significantly different from viscous fluid with a parabolic distribution. The convection heat transfer coefficient increases while the particle size of nanoparticle in nanofluids decreases. And the convection heat transfer coefficient of nanofluids is in good agreement with the experimental result both in low （0 ~〈 0.1% ） and high （ q = 0.6% ） volume fractions. In presented model, Brown motion, the effect of interactions between nanoparticles and fluid coupling, is also considered, but any phenomenological parameter is not introduced. Nanoparticles in liquid transport distribution can be quantitatively calculated by this model.

Asymptotic Behavior of Solutions for a Periodic Transport Equation with Delay

This paper analyzes a class of first order partial differential equations with delay (a model for the blood production system). The asymptotic behavior of solutions are studied.

AVERAGE REGULARITY OF THE SOLUTION TO AN EQUATION WITH THE RELATIVISTIC-FREE TRANSPORT OPERATOR

Let u = u(t, x, p) satisfy the transport equation ?u/?t+p/p0 ?u/?x= f, where f =f(t, x, p) belongs to L~p((0, T) × R~3× R~3) for 1 < p < ∞ and ?/?t+p/p0 ?/?x is the relativisticfree transport operator from the relativistic Boltzmann equation. We show the regularity of ∫u(t, x, p)d p using the same method as given by Golse, Lions, Perthame and Sentis. This average regularity is considered in terms of fractional Sobolev spaces and it is very useful for the study of the existence of the solution to the Cauchy problem on the relativistic Boltzmann equation.

A SPECTRAL STREAMLINE DIFFUSION FINITE ELEMENT COUPLING METHOD OF UNSTEADY TRANSPORT EQUATION IN THE FIELD OF NEUTRON LOGGING

In this paper, a new numerical method, the coupling method of spherical harmonic function spectral and streamline diffusion finite element for unsteady Boltzmann equation in the neutron logging field, is discussed. The convergence and error estimations of this scheme are proved. Its applications in the field of neutron logging show its effectiveness.

ON THE TRANSPORT EQUATION WITH INTEGRAL BOUNDARY CONDITIONS AND ITS CONSERVAIVE LAW

The well-posedness for the time-dependent neutron transport equationwith integral boundary conditions is established in L^1 space. Some spectral propertiesof the transport operator are discussed, the dominant eigenvalue is proved existing,and furthermore, the conservative law of migrating particle numbers is established.

An Isogeometric Error Estimate for Transport Equation in 2D

In this paper, an isogeometric error estimate for transport equation is obtained in 2D to prove the convergence of isogeometric method. The result that we have obtained, generalizes Ern result, got in finite elements method. For the time discretization, the two stage Heun scheme is used to prove this result. For a polynomial of degree k≥1, the order of convergence in space is 2 and in time is .

NONLINEAR SEMIGROUP APPROACH TOTRANSPORT EQUATIONS WITH DELAYED NEUTRONS

This paper deal with a nonlinear transport equation with delayed neutron andgeneral boundary conditions. We establish, via the nonlinear semigroups approach, the exis-tence and uniqueness of the mild solution, weak solution, strong solution and local solutionon LP-spaces （1 ≤ p 〈 ＋∞）. Local and non local evolution problems are discussed.

Analysis of anomalous transport based on radial fractional diffusion equation

Anomalous transport in magnetically confined plasmas is investigated by radial fractional transport equations.It is shown that for fractional transport models,hollow density profiles are formed and uphill transports can be observed regardless of whether the fractional diffusion coefficients(FDCs)are radially dependent or not.When a radially dependent FDC<D_(α)(r)1 is imposed,compared with the case under=D_(α)(r)1.0,it is observed that the position of the peak of the density profile is closer to the core.Further,it is found that when FDCs at the positions of source injections increase,the peak values of density profiles decrease.The non-local effect becomes significant as the order of fractional derivative a 1 and causes the uphill transport.However,as a 2,the fractional diffusion model returns to the standard model governed by Fick’s law.

Alternative constitutive relation for momentum transport of extended Navier–Stokes equations

The classical Navier–Stokes equation(NSE)is the fundamental partial differential equation that describes the flow of fluids,but in certain cases,like high local density and temperature gradient,it is inconsistent with the experimental results.Some extended Navier–Stokes equations with diffusion terms taken into consideration have been proposed.However,a consensus conclusion on the specific expression of the additional diffusion term has not been reached in the academic circle.The models adopt the form of the generalized Newtonian constitutive relation by substituting the convection velocity with a new term,or by using some analogy.In this study,a new constitutive relation for momentum transport and a momentum balance equation are obtained based on the molecular kinetic theory.The new constitutive relation preserves the symmetry of the deviation stress,and the momentum balance equation satisfies Galilean invariance.The results show that for Poiseuille flow in a circular micro-tube,self-diffusion in micro-flow needs considering even if the local density gradient is very low.

A RANDOM TRANSPORT-DIFFUSION EQUATION

In this paper we study the integral curve in a random vector field perturbed by white noise. It is related to a stochastic transport-diffusion equation. Under some conditions on the covariance function of the vector field, the solution of this stochastic partial differential equation is proved to have moments. The exact p-th moment is represented through integrals with respect to Brownian motions. The basic tool is Girsanov formula.