Regularised Finite Difference Methods for the Logarithmic Klein-Gordon Equation

Yan, Jingye

doi:10.4208/eajam.140820.250820

Cited by 4 publications

(2 citation statements)

References 44 publications

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“…Li et al [27] applied the FDM to solve the numerical solutions of the regularized LogSE in an unbounded domain. Later, for the LogKGE, two energy-conservative regularized FDMs were employed and their error estimates were obtained [48,49]. It is well known that logarithmic function will only appear numerical blow-up when ρ → 0 + , and this phenomenon will not occur when the value of ρ is large.…”

Section: Introductionmentioning

confidence: 99%

Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation

Chen,

Zhao

2024

JCM

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This paper presents three regularized models for the logarithmic Klein-Gordon equation. By using a modified Crank-Nicolson method in time and the Galerkin finite element method (FEM) in space, a fully implicit energy-conservative numerical scheme is constructed for the local energy regularized model that is regarded as the best one among the three regularized models. Then, the cut-off function technique and the time-space error splitting technique are innovatively combined to rigorously analyze the unconditionally optimal and high-accuracy convergence results of the numerical scheme without any coupling condition between the temporal step size and the spatial mesh width. The theoretical framework is uniform for the other two regularized models. Finally, numerical experiments are provided to verify our theoretical results. The analytical techniques in this work are not limited in the FEM, and can be directly extended into other numerical methods. More importantly, this work closes the gap for the unconditional error/stability analysis of the numerical methods for the logarithmic systems in higher dimensional spaces.

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Section: Introductionmentioning

confidence: 99%

Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation

Chen,

Zhao

2024

JCM

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“…There have been extensive theoretical studies on the KGD (1.1) system, including the local and global well-posedness of the Cauchy problem and the existences of bound state solutions, for which we refer to [2,14,15,19,20,23,24,32,35] and references therein. For the numerical part, different kinds of numerical methods, including the finite difference time domain (FDTD) methods and spectral methods have been proposed and analyzed for efficient computations of wave propagation in classical quantum physics, i.e., dispersive waves in the Gross-Pitaevskii equation [3], the Klein-Gordon equation [7,8,12,22,29,34,40,43], the Dirac equation [1, 4-6, 26, 28, 30, 39], the Klein-Gordon-Schrodinger equations [10,21], the Klein-Gordon-Zakharov equations [11,16,37], the Maxwell-Dirac equations [9,31], etc. However, the approaches for KGD (1.1) proposed in the literature [17,41,42] are limited.…”

Section: Introductionmentioning

confidence: 99%

Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

Bian¹,

Cheng²,

null³

et al. 2023

NMTMA

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In this paper, we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac (KGD) system. Differing from the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system, we translate the KGD equations into an equivalent system by introducing an auxiliary function, then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system. The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD, while the energy preserved by the existing conservative numerical schemes expressed by two-level's solution at each time step. By using energy method together with the 'cut-off' function technique, we establish the optimal error estimate of the numerical solution, and the convergence rate is O(τ 2 + h 2 ) in l ∞ -norm with time step τ and mesh size h. Numerical experiments are carried out to support our theoretical conclusions.

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A novel regularized model for the logarithmic Klein-Gordon equation

Yan

Zhang

et al. 2022

Applied Numerical Mathematics

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Regularised Finite Difference Methods for the Logarithmic Klein-Gordon Equation

Cited by 4 publications

References 44 publications

Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation

Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation

Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

A novel regularized model for the logarithmic Klein-Gordon equation

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