Asymptotic-Preserving Discrete Schemes for Non-Equilibrium Radiation Diffusion Problem in Spherical and Cylindrical Symmetrical Geometries

We study the asymptotic-preserving fully discrete schemes for nonequilibrium radiation diffusion problem in spherical and cylindrical symmetric geometry.The research is based on two-temperature models with Larsen’s flux-limited diffusion operators.Finite volume spatially discrete schemes are developed to circumvent the singularity at the origin and the polar axis and assure local conservation.Asymmetric second order accurate spatial approximation is utilized instead of the traditional first order one for boundary flux-limiters to consummate the schemes with higher order global consistency errors.The harmonic average approach in spherical geometry is analyzed,and its second order accuracy is demonstrated.By formal analysis,we prove these schemes and their corresponding fully discrete schemes with implicitly balanced and linearly implicit time evolutions have first order asymptoticpreserving properties.By designing associated manufactured solutions and reference solutions,we verify the desired performance of the fully discrete schemes with numerical tests,which illustrates quantitatively they are first order asymptotic-preserving and basically second order accurate,hence competent for simulations of both equilibrium and non-equilibrium radiation diffusion problems.

Review of Collocation Methods and Applications in Solving Science and Engineering Problems

The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations.This paper provides a comprehensive review of collocation methods and their applications,focused on elasticity,heat conduction,electromagnetic field analysis,and fluid dynamics.The merits of the collocation method can be attributed to the need for element mesh,simple implementation,high computational efficiency,and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method.Beginning with the fundamental principles of the collocation method,the discretization process in the continuous domain is elucidated,and how the collocation method approximation solutions for solving differential equations are explained.Delving into the historical development of the collocation methods,their earliest applications and key milestones are traced,thereby demonstrating their evolution within the realm of numerical computation.The mathematical foundations of collocation methods,encompassing the selection of interpolation functions,definition of weighting functions,and derivation of integration rules,are examined in detail,emphasizing their significance in comprehending the method’s effectiveness and stability.At last,the practical application of the collocation methods in engineering contexts is emphasized,including heat conduction simulations,electromagnetic coupled field analysis,and fluid dynamics simulations.These specific case studies can underscore collocation method’s broad applicability and effectiveness in addressing complex engineering challenges.In conclusion,this paper puts forward the future development trend of the collocation method through rigorous analysis and discussion,thereby facilitating further advancements in research and practical applications within these fields.

A Principal Theorem of Normal Discretization Schemes for Operator Equations of the First Kind

The authors announce a newly-proved theorem of theirs. This theorem is of principal significance to numerical computation of operator equations of the first kind.

A discrete unified gas-kinetic scheme for multi-species rarefied flows

A discrete unified gas kinetic scheme(DUGKS)is developed for multi-species flow in all flow regimes based on the Andries-Aoki-Perthame(AAP)kinetic model.Although the species collision operator in the AAP model conserves fully the mass,momentum,and energy for the mixture,it does not conserve the momentum and energy for each species due to the inter-species collisions.In this work,the species collision operator is decomposed into two parts:one part is fully conservative for the species and the other represents the excess part.With this decomposition,the kinetic equation is solved using the Strang-splitting method,in which the excess part of the collision operator is treated as a source,while the kinetic equation with the species conservative part is solved by the standard DUGKS.Particularly,the time integration of the source term is realized by either explicit or implicit Euler scheme.By this means,it is easy to extend the scheme to gas mixtures composed of Maxwell or hard-sphere molecules,while the previous DUGKS[Zhang Y,Zhu L,Wang R et al,Phys Rev E 97(5):053306,2018]of binary gases was only designed for Maxwell molecules.Several tests are performed to validate the scheme,including the shock structure under different Mach numbers and molar concentrations,the Couette flow under different mass ratios,and the pressure-driven Poiseuille flow in different flow regimes.The results are compared with those from other reliable numerical methods based on different models.And the influence of molecular model on the flow characteristics is studied.The results also show that the present DUGKS with implicit source discretization is more stable and preferable for gas mixture problems involving different flow regimes.

Adaptive partitioning-based discrete unified gas kinetic scheme for flows in all flow regimes

To improve the efficiency of the discrete unified gas kinetic scheme(DUGKS)in capturing cross-scale flow physics,an adaptive partitioning-based discrete unified gas kinetic scheme(ADUGKS)is developed in this work.The ADUGKS is designed from the discrete characteristic solution to the Boltzmann-BGK equation,which contains the initial distribution function and the local equilibrium state.The initial distribution function contributes to the calculation of free streaming fluxes and the local equilibrium state contributes to the calculation of equilibrium fluxes.When the contribution of the initial distribution function is negative,the local flow field can be regarded as the continuous flow and the Navier-Stokes(N-S)equations can be used to obtain the solution directly.Otherwise,the discrete distribution functions should be updated by the Boltzmann equation to capture the rarefaction effect.Given this,in the ADUGKS,the computational domain is divided into the DUGKS cell and the N-S cell based on the contribu-tion of the initial distribution function to the calculation of free streaming fluxes.In the N-S cell,the local flow field is evolved by solving the N-S equations,while in the DUGKS cell,both the discrete velocity Boltzmann equation and the correspond-ing macroscopic governing equations are solved by a modified DUGKS.Since more and more cells turn into the N-S cell with the decrease of the Knudsen number,a significant acceleration can be achieved for the ADUGKS in the continuum flow regime as compared with the DUGKS.

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model.The spatial discretization uses piecewise polynomials of degree k(k≥1)for the stress approximation,degree k+1 for the velocity approximation,and degree k for the numerical trace of velocity on the inter-element boundaries.The temporal discretization in the fully discrete method adopts a backward Euler difference scheme.We show the existence and uniqueness of the semi-discrete and fully discrete solutions,and derive optimal a priori error estimates.Numerical examples are provided to support the theoretical analysis.

Equivalent formulae of stress Green's functions for a constant slip rate on a triangular fault

We present an equivalent form of the expres- sions first obtained by Tada （Geophys J Int 164：653-669, 2006. doi： 10.1111/j. 1365-246X.2006.03868.x）, which rep- resents the transient stress response of an infinite, homo- geneous and isotropic medium to a constant slip rate on a triangular fault that continues perpetually after the slip onset. Our results are simpler than Tada＇s, and the corre- sponding codes have a higher running speed.

A Coupled Discrete Unified Gas-Kinetic Scheme for Convection Heat Transfer in Porous Media

In this paper,the discrete unified gas-kinetic scheme(DUGKS)is extended to the convection heat transfer in porous media at representative elementary volume(REV)scale,where the changes of velocity and temperature fields are described by two kinetic equations.The effects from the porous medium are incorporated into the method by including the porosity into the equilibrium distribution function,and adding a resistance force in the kinetic equation for the velocity field.The proposed method is systematically validated by several canonical cases,including the mixed convection in porous channel,the natural convection in porous cavity,and the natural convection in a cavity partially filled with porous media.The numerical results are in good agreement with the benchmark solutions and the available experimental data.It is also shown that the coupled DUGKS yields a second-order accuracy in both temporal and spatial spaces.

Pseudopotential-based discrete unified gas kinetic scheme for modeling multiphase fluid flows

To directly incorporate the intermolecular interaction effects into the discrete unified gas-kinetic scheme(DUGKS)for simulations of multiphase fluid flow,we developed a pseudopotential-based DUGKS by coupling the pseudopotential model that mimics the intermolecular interaction into DUGKS.Due to the flux reconstruction procedure,additional terms that break the isotropic requirements of the pseudopotential model will be introduced.To eliminate the influences of nonisotropic terms,the expression of equilibrium distribution functions is reformulated in a moment-based form.With the isotropy-preserving parameter appropriately tuned,the nonisotropic effects can be properly canceled out.The fundamental capabilities are validated by the flat interface test and the quiescent droplet test.It has been proved that the proposed pseudopotential-based DUGKS managed to produce and maintain isotropic interfaces.The isotropy-preserving property of pseudopotential-based DUGKS in transient conditions is further confirmed by the spinodal decomposition.Stability superiority of the pseudopotential-based DUGKS over the lattice Boltzmann method is also demonstrated by predicting the coexistence densities complying with the van der Waals equation of state.By directly incorporating the intermolecular interactions,the pseudopotential-based DUGKS offers a mesoscopic perspective of understanding multiphase behaviors,which could help gain fresh insights into multiphase fluid flow.

Pumping Schemes Impacts on SNR of Discreted Raman Amplifications

We compare different discreted DCF Raman amplifier configurations, including single-stage and dual-stage. The optimum design with respect to SNR degradation, compromise linear and nonlinear impairments.

ANALYSIS ON A NUMERICAL SCHEME WITH SECOND-ORDER TIME ACCURACY FOR NONLINEAR DIFFUSION EQUATIONS

A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It does not stir numerical oscillation,while permits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson(CN)scheme and the backward difference formula second-order(BDF2)scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.

TIME DISCRETIZATION SCHEMES FOR AN INTEGRO-DIFFERENTIAL EQUATION OF PARABOLIC TYPE

In this paper a new approach for time discretization of an integro-differential equation of parabolic type is proposed. The methods are based on the backward-Euler and Crank-Nicolson Schemes but the memory and computational requirements are greatly reduced without assuming more regularities on the solution u.

Generalized Rayleigh quotient and finite element two-grid discretization schemes

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and non-conforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in L1(L2)norm.This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes.Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.

An H^1-Galerkin Nonconforming Mixed Finite Element Method for Integro-Differential Equation of Parabolic Type

H1-Galerkin nonconforming mixed finite element methods are analyzed for integro-differential equation of parabolic type.By use of the typical characteristic of the elements,we obtain that the Galerkin mixed approximations have the same rates of convergence as in the classical mixed method,but without LBB stability condition.

A STRVCTURE-PRESERVING DISCRETIZATION OFNONLINEAR SCHRDINGER EQUATION

This paper studies the geometric structure of nonlinear Schrsdinger equationand from the view-point of preserving structure a kind of fully discrete schemes ispresented for the numerical simulation of this important equation in quantum. Ithas been shown by theoretical analysis and numerical experiments that such discrete schemes are quite satisfactory in keeping the desirable conservation propertiesand for simulating the long-time behaviour.

A NEW APPROACH TO MODELING OF TRAFFIC FLOW IN CITIES

Based on the properties of traffic, macroscopic dynamic equations are set up to simulate the flow of traffic. Besides relaxation term and compressibility term, a changeable area term is composed in the model to describe the effect of car parking, toll-booth, traffic incidents, road grading and bicycles on traffic flow. With numerical method, three cases are presented to simulate the changing lane area with time and space and the propagating process of high traffic density. The results show that the dynamic model can reasonably describe the evolution of traffic states in various complicated occasions.

High-Order Low Dissipation Conforming Finite-Element Discretization of the Maxwell Equations

In this paper,we study high order discretization methods for solving the Maxwell equations on hybrid triangle-quad meshes.We have developed high order finite edge element methods coupled with different high order time schemes and we compare results and efficiency for several schemes.We introduce in particular a class of simple high order low dissipation time schemes based on a modified Taylor expansion.

Modeling and simulation of chemically reacting flows in gas-solid catalytic and non-catalytic processes

This paper gives an overview of the recent development of modeling and simulation of chemically react- ing flows in gas-solid catalytic and non-catalytic processes. General methodology has been focused on the Eulerian-Lagrangian description of particulate flows, where the particles behave as the catalysts or the reactant materials. For the strong interaction between the transport phenomena （i.e., momentum, heat and mass transfer） and the chemical reactions at the particle scale, a cross-scale modeling approach, i.e., CFD-DEM or CFD-DPM, is established for describing a wide variety of complex reacting flows in multiphase reactors, Representative processes, including fluid catalytic cracking （FCC）, catalytic conversion of syngas to methane, and coal pyrolysis to acetylene in thermal plasma, are chosen as case studies to demonstrate the unique advantages of the theoretical scheme based on the integrated particle-scale information with clear physical meanings, This type of modeling approach provides a solid basis for understanding the multiphase reacting flow problems in general.

POINTWISE AND UPWIND DISCRETIZATIONS OF SOURCE TERMS IN OPEN-CHANNEL FLOOD ROUTING

Upwind algorithms are becoming progressively popular for river flood routing due to their capability of resolving trans-critical flow regimes. For consistency, these algorithms suggest natural upwind discretization of the source term, which may be essential for natural channels with irregular geometry. Yet applications of these upwind algorithms to natural river flows are rare, and in such applications the traditional and simpler pointwise, rather than upwind discretization of the source term is used. Within the framework of a first-order upwind algorithm, this paper presents a comparison of upwind and pointwise discretizations of the source term. Numerical simulations were carried out for a selected irregular channel comprising a pool-riffle sequence Jn the River Lune, England with observed data. It is Shown that the impact of pointwise discretization, compared to the upwind, is appreciable mainly in flow zones with the Froude number closer to or larger than unity. The discrepancy due to pointwise and upwind discretizations of the source term is negligible in flow depth and hence in water surface elevation, but well manifested in mean velocity and derived flow quantities. Also the occurrence of flow reversal and equalisation over the pool-riffle sequence in response to increasing discharges is demonstrated.