Equilibrium theory in 2D Riemann manifold for heterogeneous biomembranes with arbitrary variational modes

Based on the first and second gradient operators and their integral theorems in 2D Riemann manifold, the equilibrium differential equations and geometrically constraint equations for heterogeneous biomembranes with arbitrary variation modes are developed. Through the combination of these equations, the equilibrium theory for heterogeneous biomembranes is established in 2D Riemann manifold. From the equilibrium theory, various interesting information is revealed: Different from homogeneous biomembranes, heterogeneous one posses new equations within the membrane’s tangential planes, i.e. the in-plane equilibrium differential equations, the in-plane boundary conditions and the in-plane geometrically constraint equations. Different from the equilibrium theory in Euclidean space, the one in 2D Riemann manifold displays strict constraints between the physical coefficients and characteristic geometric parameters of biomembranes.     

The application of generalized Cole-Hopf substitutions in compressible-fluid hydrodynamics

The method for integrating nonlinear equations using generalized Cole-Hopf substitutions is extended to the 1+2 dimension. The general structure of the solutions to the Euler equations for 2D compressible-fluid flows is analyzed. A method is developed for constructing new exact solutions to describe 2D flows of compressible and incompressible fluids.     

Mach reflection for the two-dimensional Burgers equation

We study shock reflection for the two 2D Burgers equation. This model equation is an asymptotic limit of the Euler equations, and retains many of the features of the full equations. A von Neumann type analysis shows that the 2D Burgers equation has detachment, sonic, and Crocco points in complete analogy with gas dynamics. Numerical solutions support the detachment/sonic criterion for transition from regular to Mach reflection. There is also strong numerical evidence that the reflected shock in the 2D Burgers Mach reflection forms a smooth wave near the Mach stem, as proposed by Colella and Henderson in their study of the Euler equations.     

A direct Eulerian GRP scheme for relativistic hydrodynamics: Two-dimensional case

This paper develops a direct Eulerian generalized Riemann problem (GRP) scheme for two-dimensional (2D) relativistic hydrodynamics (RHD). It is an extension of the GRP scheme for one-dimensional (1D) RHDs [Z.C. Yang, P. He, H.Z. Tang, J. Comput. Phys. 230 (2011) 7964–7987] and the GRP scheme for the non-relativistic hydrodynamics [M. Ben-Artzi, J.Q. Li, G. Warnecke, J. Comput. Phys. 218 (2006) 19–43]. In order to derive the direct Eulerian GRP scheme, the (local) GRP of the split 2D RHD equations in the Eulerian formulation has to be directly resolved by using corresponding Riemann invariants and Rankine–Hugoniot jump conditions so that the crucial and delicate Lagrangian treatment in the original GRP scheme [M. Ben-Artzi, J. Falcovitz, J. Comput. Phys. 55 (1984) 1–32] may be avoided. An important difference of resolving the GRP of the split 2D RHD equations from the GRP of the 1D RHD equations or the non-relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the GRP of the split 2D RHD equations are nonlinearly coupled through the Lorentz factor which is also built in terms of the tangential velocities. It is a purely multi-dimensional relativistic feature. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2D GRP scheme.     

基于静动态重构的高阶DG/FV混合格式在二维非结构网格中的推广

通过比较间断Galerkin有限元方法(DGM)和有限体积方法(FVM),提出"静态重构"和"动态重构"的概念,进一步建立基于静动态"混合重构"算法的三阶DG/FV混合格式.在DG/FV混合格式中,单元平均值和一阶导数由DGM方法"动态重构",二阶导数利用FVM方法"静态重构";在此基础上,构造高阶多项式插值函数,得到...     

求解二维Euler方程有限单元边插值的降维重构算法

数值求解二维Euler方程的有限体积法（如k-exact，WENO重构、紧致重构等），无一例外地要进行耗时的网格单元上的二维重构.然而这些二维重构最后仅用于确定网格单元边界上高斯积分点处的解值，单元上二维重构似乎并非必需的.因此，文章提出用网格边上的一维重构来取代有限体积法中网格单元上的二维重构，分别在一致矩形网格和非结构三角形网格上发展了基于网格边重构的求解二维Euler方程的新方法，称为降维重构算法.数值算例表明该算法可以计算有强激波的无黏流动问题，且有较高的计算效率.      

Closed-form analytical solutions of the time difference of arrival source location problem for minimal element monitoring arrays

Closed-form analytical solutions are found for the time difference of arrival (TDOA) source location problem. Solutions are found for both two-dimensional (2D) and three-dimensional (3D) source location by formulating the TDOA equations in, respectively, polar and spherical coordinate systems, with the radial direction coincident with the assumed geodesic path of signal propagation to a reference sensor. Quadratic equations for TDOA 2D and 3D source location based on the spherical intersection (SX) scheme, in some cases permitting dual physical solutions, are found for three and four sensor element monitoring arrays, respectively. A method of spherical intersection subarrays (SXSAs) is developed to derive from these quadratic equations globally unique closed-form analytical solutions for TDOA 2D and 3D source location, for four and five sensor element monitoring arrays, respectively. Errors in 2D source location for introduced bias in time differences of arrival are shown to have a strong geometrical dependence. The SXSA and SX methods perform well in terms of accuracy and precision at high levels of arrival time bias for both 2D and 3D source location and are much more efficient than nonlinear least-squares schemes. The SXSA scheme may have particular applicability to accurately solving source location problems in demanding real-time situations.     

Rigorous Remarks about Scaling Laws in Turbulent Fluids

A definition of scaling law for suitable families of measures is given and investigated. First, a number of necessary conditions are proved. They imply the absence of scaling laws for 2D stochastic Navier-Stokes equations and for the stochastic Stokes (linear) problem in any dimension, while they imply a lower bound on the mean vortex stretching in 3D. Second, for the 3D stochastic Navier-Stokes equations, necessary and sufficient conditions for scaling laws to hold are given, translating the problem into bounds for energy and enstrophy of high and low modes respectively. Unlike in the 2D case, the validity or invalidity of such conditions in 3D remains open.     

Implicit DG Method for Time Domain Maxwell's Equations Involving Metamaterials

An implicit discontinuous Galerkin method is introduced to solve the time-domain Maxwell's equations in metamaterials. The Maxwell's equations in metamaterials are represented by integral-differential equations. Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain. The fully discrete numerical scheme is proved to be unconditionally stable. When polynomial of degree at most $p$ is used for spatial approximation, our scheme is verified to converge at a rate of $\mathcal{O}(τ^2+h^{p+1/2})$. Numerical results in both 2D and 3D are provided to validate our theoretical prediction.     

Direct quadrature method of moments solution of the Fokker–Planck equation

The direct quadrature method of moments is presented as an efficient and accurate means of numerically computing solutions of the Fokker–Planck equation corresponding to stochastic nonlinear dynamical systems. The theoretical details of the solution procedure are first presented. The method is then used to solve Fokker–Planck equations for both 1D and 2D (noisy van der Pol oscillator) processes which possess nonlinear stochastic differential equations. Higher-order moments of the stationary solutions are computed and prove to be very accurate when compared to analytic (1D process) and Monte Carlo (2D process) solutions.     

二维静电场E分量的直接解法

本文从静电场的基本方程出发，由基于矩阵算子的分离变量法直接求二维场域中各点的电场强度Ｅ的大小和方向，而不引入辅助势．此项工作拓宽了分离变量法的应用范围，同时，给出了静电场问题的一种新解法     

Turbulence of incompressible liquid flow near local equilibrium and the principle of secondary maximum of entropy

The variational principle of maximum entropy is used to describe the dynamics of weakly nonequilibrium turbulence using the theory of Reynolds stresses for viscous incompressible liquid flow. From this principle, equations closing the theory of Reynolds stresses and also equations describing mean flow-turbulence interaction for 3D turbulent flows are derived. The theory is reduced to 2D flows and weak turbulence. Thermodynamic analogues and an example of Couette flow are considered.     

Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.     

Integrating 1D and 2D hydrodynamic,sediment transport model for dam-break flow using finite volume method

The purpose of this study is to set up a dynamically linked 1D and 2D hydrodynamic and sediment transport models for dam break flow.The 1D-2D coupling model solves the generalized shallow water equations,the non-equilibrium sediment transport and bed change equations in a coupled fashion using an explicit finite volume method.It considers interactions among transient flow,strong sediment transport and rapid bed change by including bed change and variable flow density in the flow continuity and momentum equations.An unstructured Quadtree rectangular grid with local refinement is used in the 2D model.The intercell flux is computed by the HLL approximate Riemann solver with shock captured capability for computing the dry-to-wet interface for all models.The effects of pressure and gravity are included in source term in this coupling model which can simplify the computation and eliminate numerical imbalance between source and flux terms.The developed model has been tested against experimental and real-life case of dam-break flow over fix bed and movable bed.The results are compared with analytical solution and measured data with good agreement.The simulation results demonstrate that the coupling model is capable of calculating the flow,erosion and deposition for dam break flows in complicated natural domains.     

Möbius invariant integrable lattice equations associated with KP and 2DTL hierarchies

The integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. They generate the corresponding continuous hierarchy of singular manifold equations, its Bäcklund transformations and different forms of superposition principles; their distinctive feature is invariance under the action of Möbius transformation. Geometric interpretation of these discrete equations is given.     

Formation of a vortex lattice in a rotating BCS Fermi gas

We investigate theoretically the formation of a vortex lattice in a superfluid two-spin component Fermi gas in a rotating harmonic trap, in a BCS-type regime of condensed non-bosonic pairs. Our analytical solution of the superfluid hydrodynamic equations, both for the 2D BCS equation of state and for the 3D unitary quantum gas, predicts that the vortex free gas is subject to a dynamic instability for fast enough rotation. With a numerical solution of the full time dependent BCS equations in a 2D model, we confirm the existence of this dynamic instability and we show that it leads to the formation of a regular pattern of quantum vortices in the gas.     

3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis

We prove by an explicit construction that solutions to incompressible 3D Euler equations defined in the periodic cube Ω=[0,L]³ can be mapped bijectively to a new system of equations whose solutions are globally regular. We establish that the usual Beale-Kato-Majda criterion for finite-time singularity (or blowup) of a solution to the 3D Euler system is equivalent to a condition on the corresponding regular solution of the new system. In the hypothetical case of Euler finite-time singularity, we provide an explicit formula for the blowup time in terms of the regular solution of the new system. The new system is amenable to being integrated numerically using similar methods as in Euler equations. We propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite-time singularity problem of Euler equations, based on the conservation of quantities directly related to energy and circulation. The method of mapping to a regular system can be extended to any fluid equation that admits a Beale-Kato-Majda type of theorem, e.g. 3D Navier-Stokes, 2D and 3D magnetohydrodynamics, and 1D inviscid Burgers. We discuss briefly the case of 2D ideal magnetohydrodynamics. In order to illustrate the usefulness of the mapping, we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1D inviscid Burgers equation.     

On the Self-Similar Solutions of the 3D Euler and the Related Equations

We generalize and localize the previous results by the author on the study of self-similar singularities for the 3D Euler equations. More specifically we extend the restriction theorem for the representation for the vorticity of the Euler equations in a bounded domain, and localize the results on asymptotically self-similar singularities. We also present progress towards relaxation of the decay condition near infinity for the vorticity of the blow-up profile to exclude self-similar blow-ups. The case of the generalized Navier-Stokes equations having the laplacian with fractional powers is also studied. We apply the similar arguments to the other incompressible flows, e.g. the surface quasi-geostrophic equations and the 2D Boussinesq system both in the inviscid and supercritical viscous cases.     

Statistical equilibrium solutions of the shallow water equations

A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2D fluid flow with a free surface, is described. The model contains a competing acoustic turbulent direct energy cascade, and a 2D turbulent inverse energy cascade. It is shown, nonetheless that, just as in the corresponding theory of the inviscid Euler equation, the infinite number of conserved quantities constrains the flow sufficiently to produce nontrivial large-scale vortex structures which are solutions to a set of explicitly derived coupled nonlinear partial differential equations.