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Numerical Validation for High Order Hyperbolic Moment System of wigner equation

Communications in Computational Physics 2014年第3期15卷 569-595页

作者： Ruo Li Tiao Lu Yanli Wang Wenqi Yao School of Mathematical Sciences Peking UniversityBeijing 100871P.R.China HEDPS and LMAM Peking UniversityBeijing 100871P.R.China BICMR Peking UniversityBeijing 100871P.R.China CAPT Peking UniversityBeijing 100871P.R.China

A globally hyperbolic moment system upto arbitrary order for the wigner equation was derived in[6].For numerically solving the high order hyperbolic moment system therein,we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in[8],to validate the moment system of the wigner equation.The method developed can keep both mass and momentum conserved,and the variation of the total energy under control though it is not strictly conservative.We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion,and the convergence of the numerical solution of the moment system to the numerical solution of the wigner equation using the discrete velocity method.The numerical results indicate that the high order moment system in[6]is a valid model for the wigner equation,and the proposed numerical method for the moment system is quite promising to carry out the simulation of the wigner equation.

关键词： wigner equation NRxx method moment method

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Adaptive Conservative Cell Average Spectral Element Methods for Transient wigner equation in Quantum Transport

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Communications in Computational Physics 2011年第3期9卷 711-739页

作者： Sihong Shao Tiao Lu Wei Cai LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Department of Mathematics and Statistics University of North Carolina at CharlotteCharlotteNC 28223USA

A new adaptive cell average spectral element method(SEM)is proposed to solve the time-dependent wigner equation for transport in quantum devices.The proposed cell average SEM allows adaptive non-uniform meshes in phase spaces to reduce the high-dimensional computational cost of wigner functions while preserving exactly the mass conservation for the numerical solutions.The key feature of the proposed method is an analytical relation between the cell averages of the wigner function in the k-space(local electron density for finite range velocity)and the point values of the distribution,resulting in fast transforms between the local electron density and local fluxes of the discretized wigner equation via the fast sine and cosine transforms.Numerical results with the proposed method are provided to demonstrate its high accuracy,conservation,convergence and a reduction of the cost using adaptive meshes.

关键词： wigner equation quantum transport spectral element methods adaptive mesh

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A Branching Random Walk Method for Many-Body wigner Quantum Dynamics

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Numerical Mathematics(Theory,Methods and Applications) 2019年第1期12卷 21-71页

作者： Sihong Shao Yunfeng Xiong LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China

A branching random walk algorithm for many-body wigner equations and its numerical applications for quantum dynamics in phase space are proposed and ana-lyzed in this paper.Using an auxiliary function,the truncated wigner equation and its adjoint form are cast into integral formulations,which can be then reformulated into renewal-type equations with probabilistic interpretations.We prove that the first mo-ment of a branching random walk is the solution for the adjoint equation.With the help of the additional degree of freedom offered by the auxiliary function,we are able to produce a weighted-particle implementation of the branching random walk.In contrast to existing signed-particle implementations,this weighted-particle one shows a key ca-pacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors.Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings,and demonstrate the accuracy,the efficiency,and thus the computability of the proposed weighted-particle wigner branching random walk algorithm.

关键词： wigner equation branching random walk quantum dynamics variance reduction signed-particle Monte Carlo method adjoint equation renewal-type equations importance sam-pling resampling

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