Localized space–time method of fundamental solutions for three-dimensional transient diffusion problem

Qiu, Lin; Lin, Ji; Qin, Qi; Chen, W.

doi:10.1007/s10409-020-00979-8

Cited by 9 publications

(3 citation statements)

References 14 publications

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“…For the one-dimensional (1D) time-dependent problem, the ST approach defines a two-dimensional (2D) steady-state problem in an x-t field. Since the ST approach is applied, the distribution nodes are set both in the space-and time-axes and named ST-domain [38,40,41], as shown in Figure 1a.…”

Section: Space-time Generalized Finite Difference Methodsmentioning

confidence: 99%

“…For the one-dimensional (1D) time-dependent problem, t approach defines a two-dimensional (2D) steady-state problem in an x-t field. Sinc ST approach is applied, the distribution nodes are set both in the space-and time-axe named ST-domain [38,40,41], as shown in Figure 1a. A supporting domain is set up by choosing the n s nearest nodes within the central i node in Figure 1b (see the "×" symbol), and (x i , t i ), i = 1, 2, .…”

Section: Space-time Generalized Finite Difference Methodsmentioning

confidence: 99%

“…This simplification enables a clearer determination of the properties of the numerical scheme, such as accuracy and stability. Recently, the ST coupled approach has been widely combined with meshless methods such as the ST localized RBFCM [38], ST kernel-based method [39], ST localized method of fundamental solutions [40,41], ST Trefftz Method [42,43], ST backward substitution method [44], and ST-GFDM [45][46][47][48][49][50][51][52][53][54]. Based on the flexibility of the ST-GFDM, researchers have applied this meshless method for engineering problems in the past few years.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space–Time Coupled Generalized Finite Difference Scheme

Zhang

2023

Applied Sciences

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This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme.

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Section: Space-time Generalized Finite Difference Methodsmentioning

confidence: 99%

Section: Space-time Generalized Finite Difference Methodsmentioning

confidence: 99%