A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

Qu, Wenzhen; Gu, Yan; Zhang, Yaoming; Fan, Chia‐Ming; Zhang, Chuanzeng

doi:10.1002/nme.5948

Cited by 36 publications

(9 citation statements)

References 54 publications

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“…In the strong form GFDM [21,22,23,24,25], for a linear differential operator D * , we seek a numerical representation D * i of that differential operator at point x i given by coefficients…”

Section: Generalized Finite Difference Methodsmentioning

confidence: 99%

Meshfree Collocation for Elliptic Problems with Discontinuous Coefficients

Kraus,

Kuhnert,

Meister

et al. 2022

Preprint

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We present a meshfree generalized finite difference method (GFDM) for solving Poisson's equation with coefficients containing jump discontinuities up to several orders of magnitude. To discretize the diffusion operator, we formulate a strong form method that uses a smearing of the discontinuity, and a conservative formulation based on locally computed Voronoi cells. Additionally, we propose a novel conservative formulation of enforcing Neumann boundary conditions that is compatible with the conservative formulation of the diffusion operator. Finally, we introduce a way to switch between the strong form and the conservative formulation to obtain a locally conservative and positivity preserving scheme. The presented numerical methods are benchmarked against four test cases with varying complexity and different jump magnitudes on point clouds that are not aligned to the discontinuity. Our results show that the new hybrid method that switches between the two formulations produces better results than the standard GFDM approach for high jumps in the diffusivity parameter.

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Section: Generalized Finite Difference Methodsmentioning

confidence: 99%

Meshfree Collocation for Elliptic Problems with Discontinuous Coefficients

Kraus,

Kuhnert,

Meister

et al. 2022

Preprint

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“…Because the Taylor series approximation becomes more accurate when the distance λ j is smaller, which should have a higher weight κ j in residual function R(H) of Equation (9). Some other weighting functions can be found in [53,59].…”

Section: Spatial Discretization By the Gfdmmentioning

confidence: 99%

“…Thanks to this spare system, this method is highly efficient and suitable for the numerical simulations of large-scale problems. Many physical applications have been addressed by the GFDM, such as the thin elastic plate bending analysis [49], the electroelastic analysis of 3D piezoelectric structures [50], the acoustic wave propagation [51], the inverse Cauchy problem in 2D elasticity [52], the heat conduction problems [53], and the stationary flow in a dam [54].…”

Section: Introductionmentioning

confidence: 99%

A Hybrid Localized Meshless Method for the Solution of Transient Groundwater Flow in Two Dimensions

Wang

Kim

2022

Mathematics

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In this work, a hybrid localized meshless method is developed for solving transient groundwater flow in two dimensions by combining the Crank–Nicolson scheme and the generalized finite difference method (GFDM). As the first step, the temporal discretization of the transient groundwater flow equation is based on the Crank–Nicolson scheme. A boundary value problem in space with the Dirichlet or mixed boundary condition is then formed at each time node, which is simulated by introducing the GFDM. The proposed algorithm is truly meshless and easy to program. Four linear or nonlinear numerical examples, including ones with complicated geometry domains, are provided to verify the performance of the developed approach, and the results illustrate the good accuracy and convergency of the method.

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“…Ure na et al [23] applied the generalized finite difference method (GFDM) to the two-dimensional (2D) seismic wave propagation problem. The GFDM, as a very powerful meshless method, has been widely used for many applications [24][25][26][27][28][29][30][31] because of its simplicity, stability and easy implementation.…”

Section: Introductionmentioning

confidence: 99%

Integrating Krylov Deferred Correction and Generalized Finite Difference Methods for Dynamic Simulations of Wave Propagation Phenomena in Long-Time Intervals

sci¹

2021

AAMM

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In this paper, a high-accuracy numerical scheme is developed for long-time dynamic simulations of 2D and 3D wave propagation phenomena. In the derivation of the present approach, the second order time derivative of the physical quantity in the wave equation is treated as a substitution variable. Based on the temporal discretization with the Krylov deferred correction (KDC) technique, the original wave problem is then converted into the modified Helmholtz equation. The transformed boundary value problem (BVP) in space is efficiently simulated by using the meshless generalized finite difference method (GFDM) with Taylor series after truncating the second and fourth order approximations. The developed scheme is finally verified by four numerical experiments including cases with complicated domains or the temporally piecewise defined source function. Numerical results match with the analytical solutions and results by the COMSOL software, which demonstrates that the developed KDC-GFDM can allow large time-step sizes for wave propagation problems in longtime intervals.

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A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

Cited by 36 publications

References 54 publications

Meshfree Collocation for Elliptic Problems with Discontinuous Coefficients

Meshfree Collocation for Elliptic Problems with Discontinuous Coefficients

A Hybrid Localized Meshless Method for the Solution of Transient Groundwater Flow in Two Dimensions

Integrating Krylov Deferred Correction and Generalized Finite Difference Methods for Dynamic Simulations of Wave Propagation Phenomena in Long-Time Intervals

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