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Semi-Eulerian and high order Gaussian Beam methods for the Schrödinger Equation in the Semiclassical Regime

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Communications in Computational Physics 2011年第3期9卷 668-687页

作者： Shi Jin Hao Wu Xu Yang Department of Mathematics University of WisconsinMadisonWI 53706USA Department of Mathematical Sciences Tsinghua UniversityBeijing 10084China Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA

A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical *** this paper,we introduce an efficient semi-Eulerian implementation of this *** new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing *** difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these *** yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase *** a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam ***,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this *** numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

关键词： Schrödinger equation semi-Eulerian Gaussian beam method high order methods

high order Deep Neural Network for Solving high Frequency Partial Differential Equations

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Communications in Computational Physics 2022年第2期31卷 370-397页

作者： Zhipeng Chang Ke Li Xiufen Zou Xueshuang Xiang Qian Xuesen Laboratory of Space Technology China Academy of Space TechnologyBeijing 100875P.R.China School of Mathematics and Statistics Wuhan UniversityWuhan 430072P.R.China Information Engineering University Zhengzhou 450001P.R.China

This paper proposes a high order deep neural network(HorderDNN)for solving high frequency partial differential equations(PDEs),which incorporates the idea of“high order”from finite element methods(FEMs)into commonly-used deep neural networks(DNNs)to obtain greater approximation *** main idea of HorderDNN is introducing a nonlinear transformation layer between the input layer and the first hidden layer to form a high order polynomial space with the degree not exceeding p,followed by a normal *** order p can be guided by the regularity of solutions of *** performance of HorderDNNis evaluated on high frequency function fitting problems and high frequency Poisson and Helmholtz *** results demonstrate that:HorderDNNs(p>1)can efficiently capture the high frequency information in target functions;and when compared to physics-informed neural network(PINN),HorderDNNs(p>1)converge faster and achieve much smaller relative errors with same number of trainable *** particular,when solving the high frequency Helmholtz equation in Section 3.5,the relative error of PINN stays around 1 with its depth and width increase,while the relative error can be reduced to around 0.02 as p increases(see Table 5).

关键词： Deep neural network high order methods high frequency PDEs

high order Deep Domain Decomposition Method for Solving high Frequency Interface Problems

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Advances in Applied Mathematics and Mechanics 2023年第6期15卷 1602-1630页

作者： Zhipeng Chang Ke Li Xiufen Zou Xueshuang Xiang School of Mathematics and Statistics Wuhan UniversityWuhanHubei 430072China Qian Xuesen Laboratory of Space Technology China Academy of Space TechnologyBeijing 100875China Information Engineering University ZhengzhouHenan 450001China

This paper proposes a high order deep domain decomposition method(HorderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HorderDNN)with domain decomposition method(DDM).The main idea of HorderDeepDDM is to divide the computational domain into some sub-domains by DDM,and apply HorderDNNs to solve the high-frequency problem on each ***,we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains,on the interface and the boundary during the optimization *** performance of HorderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface *** results indicate that:HorderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HorderDNN to approximate high-frequency *** detail,HorderDeepDDMs(p>1)could capture the high-frequency information very *** compared to the deep domain decomposition method(DeepDDM),HorderDeepDDMs(p>1)converge faster and achieve much smaller relative errors with the same number of trainable *** example,when solving the high-frequency interface elliptic problems in Section 3.3.1,the minimum relative errors obtained by HorderDeepDDMs(p=9)are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same,as shown in Fig.4.

关键词： Deep neural network high order methods high-frequency interface problems do-main decomposition method

Fourth order Schemes for Time-Harmonic Wave Equations with Discontinuous Coefficients

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Communications in Computational Physics 2009年第2期5卷 442-455页

作者： Guy Baruch Gadi Fibich Semyon Tsynkov Eli Turkel School of Mathematical Sciences Tel Aviv UniversityRamat AvivTel Aviv 69978Israel Department of Mathematics North Carolina State UniversityBox 8205RaleighNC 27695USA

We consider high order methods for the one-dimensional Helmholtz equation and frequency-Maxwell *** demand that the scheme be higher order even when the coefficients are *** discuss the connection between schemes for the second-order scalar Helmholtz equation and the first-order system for the electromagnetic or acoustic applications.

关键词： Helmholtz equation Maxwell’s equations finite volume approximation compact schemes high order methods material discontinuities

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

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Advances in Aerodynamics 2019年第1期1卷 39-74页

作者： Jiequan Li Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijingPeople’s Republic of China Center for Applied Physics and Technology Peking UniversityBeijingPeople’s Republic of China

With increasing engineering demands,there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct“physics”.There are two families of high order methods:One is the method of line,relying on the Runge-Kutta(R-K)*** building block is the Riemann solution labeled as the solution element“1”.Each step in R-K just has first order *** order to derive a fourth order accuracy scheme in time,one needs four stages labeled as“1111=4”.The other is the one-stage Lax-Wendroff(LW)type method,which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear *** recent years,the pair of solution element and dynamics element,labeled as“2”,are taken as the building *** direct adoption of the dynamics implies the inherent temporal-spatial *** this type of building blocks,a family of two-stage fourth order accurate schemes,labeled as“22=4”,are designed for the computation of compressible fluid *** resulting schemes are compact,robust and *** paper contributes to elucidate how and why high order accurate schemes should be so *** some extent,the“22=4”algorithm extracts the advantages of the method of line and one-stage LW *** a core part,the pair“2”is expounded and LW solver is *** generalized Riemann problem(GRP)solver,as the discontinuous and nonlinear version of LW flow solver,and the gas kinetic scheme(GKS)solver,the microscopic LW solver,are all *** compact Hermite-type data reconstruction and high order approximation of boundary conditions are ***,the computational performance and prospective discussions are presented.

关键词： Compressible fluid dynamics Hyperbolic balance laws high order methods Temporal-spatial coupling Multi-stage two-derivative methods Lax-Wendroff type flow solvers GRP solver