Schrödinger方程的连续时空有限元方法

本文讨论Schr?dinger方程的连续时空有限元方法,通过引入相应的时空投影算子,利用实部虚部分离技巧,得到了变量u在时间节点处的L²范数,以及u和u_t的全局L²(H¹)和L²(L²)范数意义下的最优误差估计结果.该文的结论对进一步探索和设计Schr?dinger方程的数值算法是有益的.     

Quantum-Gravitational Diffusion and Stochastic Fluctuations in the Velocity of Light

We argue that quantum-gravitational fluctuations in the space-time background give the vacuum non-trivial optical properties that include diffusion and consequent uncertainties in the arrival times of photons, causing stochastic fluctuations in the velocity of light in vacuo. Our proposal is motivated within a Liouville string formulation of quantum gravity that also suggests a frequency-dependent refractive index of the particle vacuum. We construct an explicit realization by treating photon propagation through quantum excitations of D-brane fluctuations in the space-time foam. These are described by higher-genus string effects, that lead to stochastic fluctuations in couplings, and hence in the velocity of light. We discuss the possibilities of constraining or measuring photon diffusion in vacuo via -ray observations of distant astrophysical sources.     

Review: Gas of Wormholes: A Possible Ground State of Quantum Gravity

In order to gain insight into the possible Ground State of Quantized Einstein's Gravity, we have derived a variational calculation of the energy of the quantum gravitational field in an open space, as measured by an asymptotic observer living in an asymptotically flat space-time. We find that for Quantum Gravity (QG) it is energetically favourable to perform its quantum fluctuations not upon flat space-time but around a "gas" of wormholes of mass m _p, the Planck mass (m _p 10¹⁹ GeV) and average distance l _p, the Planck length a _p(a _p 10^–33 cm). As a result, assuming such configuration to be a good approximation to the true Ground State of Quantum Gravity, space-time, the arena of physical reality, turns out to be well described by Wheeler's quantum foam and adequately modeled by a space-time lattice with lattice constant l _p, the Planck lattice.     

Local time-stepping procedures for the space-time conservation element and solution element method

A local time-stepping procedure for the space-time conservation element and solution element (CESE) method has been developed. This new procedure allows for variation of time-step size in both space and time, and can also be extended to become multi-dimensional solvers with structured/unstructured spatial grids. Moreover, it differs substantially in concept and methodology from the existing approaches. By taking full advantage of key concepts of the CESE method, in a simple and efficient manner it can enforce flux conservation across an interface separating grid zones of different time-step sizes. In particular, no correction pass is needed. Numerical experiments show that, for a variety of flow problems involving moving shock and flame discontinuities, accurate and robust numerical simulations can be achieved even with a reduction in time-step size on the order of 10 or higher for grids across a single interface.     

Formalization of the “Twin Paradox” for Non-uniformly Accelerated Motions

The use of hyperbolic numbers for studying space-time geometry and trigonometry is extended for demonstrating the Frenet’s formulas in space-time. By means of this introduction the twin paradox for non-uniformly accelerated motions is formalized in a straightforward way.     

Evaluation of time correlation functions from a generalized Enskog equation

Numerical results for the density and current correlation functions in dense hard-sphere fluids are obtained from a kinetic equation which is the extension of the linearized Enskog equation to finite wavelengths in order to demonstrate the convergence of the method of solution. Comparison is made to a previously proposed approximate solution.This work was performed in part under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG-48 and in part supported by the National Science Foundation.     

光栅面不平行对压缩脉冲时空特性的影响

 推导了光栅对压缩器两光栅在垂直于刻线的平面(衍射面)内有一夹角时1至4阶色散变化量的表达式,以及脉冲超高斯光束经光栅对压缩后的场分布。并据此模拟分析了光栅表面平行性失调对输出脉冲时空特性的影响。结果表明：脉冲光束单次通过光栅对时其波形会产生扭曲,光栅对的失调会使脉冲扭曲更加严重;而两次通过光栅对时横向谱移动的影响会消除,但脉冲将出现旁瓣,光栅对的失调使得脉冲在时间上提前,且色散阶数越高色散变化量对压缩脉冲时空特性的影响越大。     

Numerical simulation of oxide nanoparticle growth characteristics under the gas detonation chemical reaction by space-time conservation element–solution element method

Under harsh conditions (such as high temperature, high pressure, and millisecond lifetime chemical reaction), a long-standing challenge remains to accurately predict the growth characteristics of nanosize spherical particles and to determine the rapid chemical reaction flow field characteristics. The growth characteristics of similar spherical oxide nanoparticles are further studied by successfully introducing the space-time conservation element–solution element (CE/SE) algorithm with the monodisperse Kruis model. This approach overcomes the nanosize particle rapid growth limit set and successfully captures the characteristics of the rapid gaseous chemical reaction process. The results show that this approach quantitatively captures the characteristics of the rapid chemical reaction, nanosize particle growth and size distribution. To reveal the growth mechanism for numerous types of oxide nanoparticles, it is very important to choose a rational numerical method and particle physics model.     

Infinite Sequence Solutions for Space-Time Fractional Symmetric Regularized Long Wave Equation

In this paper, we investigate the space-time fractional symmetric regularized long wave equation. By using the Bäcklund transformations and nonlinear superposition formulas of solutions to Riccati equation, we present infinite sequence solutions for space-time fractional symmetric regularized long wave equation. This method can be extended to solve other nonlinear fractional partial differential equations.     

Numerical modeling of 1D transient poroelastic waves in the low-frequency range

Propagation of transient mechanical waves in porous media is numerically investigated in 1D. The framework is the linear Biot model with frequency-independent coefficients. The coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky. A method combining three numerical tools is proposed: a fourth-order ADER scheme with time-splitting to deal with the time-marching, a space-time mesh refinement to account for the small-scale evolution of the slow wave, and an interface method to enforce the jump conditions at interfaces. Comparisons with analytical solutions confirm the validity of this approach.