Exponential splitting for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math>-dimensional paraxial Helmholtz equation with high wavenumbers

Sheng, Qin; Sun, Hai‐Wei

doi:10.1016/j.camwa.2014.09.005

Cited by 5 publications

(2 citation statements)

References 17 publications

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“…Romão et al (2011) constructed two kinds of numerical method by Galerkin and least square finite element methods for solving the three-dimensional Poisson and Helmholtz equations which represent heat diffusion in solids. Sheng and Sun (2014) discussed the application of an exponential splitting method, and the method can solve the n-dimensional paraxial Helmholtz equation with highly oscillatory condition, which introduced an eikonal transformation for the decomposition of oscillation-free platforms and matrix operators. A Taylor expansion based on a fast multipole method by Wang et al (2020) was developed for implementing the low-frequency three-dimensional Helmholtz Green's functions in layered media.…”

Section: Introductionmentioning

confidence: 99%

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

Yang

2021

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PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

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Section: Introductionmentioning

confidence: 99%

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

Yang

2021

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“…where α and β are constants. The nonlinear Schrödinger equation has been extensively studied by various numerical methods, such as finite element methods, [27] finite difference methods, [28] spectral methods, [29,30] splitting methods, [6,7,18,19,[31][32][33] etc. Among these numerical methods of different categories, the multisymplectic method has attracted special attention for its better numerical stability for long-time computations and perfect performance in preserving the intrinsic properties and conservation laws of nonlinear Schrödinger equations.…”

Section: Introductionmentioning

confidence: 99%

Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

et al. 2016

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In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.

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A unified derivation of finite‐difference schemes from solution matching

Chin

2015

Numerical Methods Partial

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Conventional finite‐difference schemes for solving partial differential equations are based on approximating derivatives by finite‐differences. In this work, an alternative method is proposed which views finite‐difference schemes as systematic ways of matching up to the operator solution of the partial differential equation. By completely abandoning the idea of approximating derivatives directly, the method provides a unified description of explicit finite‐difference schemes for solving a general linear partial differential equation with constant coefficients to any time‐marching order. As a result, the stability of the first‐order algorithm for an entire class of linear equations can be determined all at once. Because the method is based on solution‐matching, it can also be used to derive any order schemes for solving the general nonlinear advection equation. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 243–265, 2016

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Exponential splitting for n-dimensional paraxial Helmholtz equation with high wavenumbers

Cited by 5 publications

References 17 publications

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

A unified derivation of finite‐difference schemes from solution matching

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