Numerical solution of nonlinear Klein–Gordon equation using the element-free kp-Ritz method

Guo, Peng; Liew, K.M.; Zhu, Ping

doi:10.1016/j.apm.2014.11.025

Cited by 8 publications

(2 citation statements)

References 47 publications

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“…The abovementioned space-time decoupled formulations normally entail costly computational resources compared with space-time coupled formulations [9][10][11][12][13][14][15]. However, in these space-time decoupled formulations, the matrices generated in the spatial discretization of nonlinear terms are normally dependent on the time-dependent unknown vector [14,15,[21][22][23]. As a result, these matrixes have to be recalculated at each time step, thereby consuming a lot of computations.…”

Section: Introductionmentioning

confidence: 99%

“…As a result, these matrixes have to be recalculated at each time step, thereby consuming a lot of computations. Since the recalculation of matrices representing the spatial discretization of nonlinear terms actually re-performs the spatial discretization at each time step, the decoupling between spatial and temporal discretization in these spacetime decoupled methods is incomplete [23]. To remove this shortcoming, Liu et al [24,25] proposed a space-time fully decoupled wavelet Galerkin method to solve nonlinear wave problems and the two-dimensional (2-D) Burgers' equations.…”

Section: Introductionmentioning

confidence: 99%

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A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

et al. 2021

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A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

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