Application of the generalized finite difference method to solve the advection–diffusion equation

Prieto, Francisco Ureña; Múñoz, José Manuel; Corvinos, Luis Gavete

doi:10.1016/j.cam.2010.05.026

Cited by 86 publications

(21 citation statements)

References 5 publications

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“…D n (x,y). Figure 4 shows the locations of the detonation shock front at times t=1, 5,9,13,17, 21 s, respectively, obtained by the simulation of the level set equation together with the DSD theory with D n =(8.066.8) mm/s, which is cited from references [4,5] and is derived from the reactive compressive Euler equations with a simplified rate law of r=2.5147(1) 1/2 s 1 , where  is the reaction progress variable. One can find that the shock fronts are curved and accelerated at the top edge of the converging part because of the negative curvature there.…”

Section: Converging Channel Problemmentioning

confidence: 99%

“…Recently, Gavete et al [11,12] made a comprehensive study of GFDM and compared the numerical precision of GFDM with that of element free Galerkin method, and the convergence, the truncation errors over irregular grids and the stability criterion of the method for parabolic and hyperbolic equations were investigated there. Prieto et al [13] described how GFDM could be applied for solving the advection-diffusion equation, as well as its corresponding analysis. The related research reports above show that GFDM is good at numerically solving partial differential equations for irregular grids in complicated areas.…”

Section: Introductionmentioning

confidence: 99%

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Numerical study of detonation shock dynamics using generalized finite difference method

Chen

Huang

2011

Sci. China Phys. Mech. Astron.

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The generalized finite difference method (GFDM) used for irregular grids is first introduced into the numerical study of the level set equation, which is coupled with the theory of detonation shock dynamics (DSD) describing the propagation of the detonation shock front. The numerical results of a rate-stick problem, a converging channel problem and an arc channel problem for specified boundaries show that GFDM is effective on solving the level set equation in the irregular geometrical domain. The arrival time and the normal velocity distribution of the detonation shock front of these problems can then be obtained conveniently with this method. The numerical results also confirm that when there is a curvature effect, the theory of DSD must be considered for the propagation of detonation shock surface, while classic Huygens construction is not suitable any more.generalized finite difference method, detonation shock dynamics, level set equation, propagation of detonation shock front, irregular grids

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Section: Converging Channel Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical study of detonation shock dynamics using generalized finite difference method

Chen

Huang

2011

Sci. China Phys. Mech. Astron.

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“…Some of the recent research work on the advection-diffusion equations can be seen in [14,[32][33][34] and related references therein. We won't introduce advection-diffusion equations in depth mainly because of the fact the generalized fractional advectiondiffusion equation is studied in this paper.…”

Section: Generalized Fractional Advectiondiffusion Equationmentioning

confidence: 99%

Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

Agrawal

2013

Open Physics

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Abstract:In this paper we study a class of new Generalized Fractional Advection-Diffusion Equations (GFADEs) with a new Generalized Fractional Derivative (GFD) proposed last year. The new GFD is defined in the Caputo sense using a weight function and a scale function. The GFADE is discussed in a bounded domain, and numerical solutions for two examples consisting of a linear and a nonlinear GFADE are obtained using an implicit finite difference approach. The stability of the numerical scheme is investigated, and the order of convergence is estimated numerically. Numerical results illustrate that the finite difference scheme is simple and effective for solving the GFADEs. We investigate the influence of weight and scale functions on the diffusion of GFADEs. Linear and nonlinear stretching and contracting functions are considered. It is found that an increasing weight function increases the rate of diffusion, and a scale function can stretch or contract the diffusion on the time domain. PACS

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“…In the recent years, the GFDM has been applied with different purposes, to solve the wave equation, to solve the advecion‐diffusion equation, to solve nonlinear partial differential equations, to solve the electrical conductivity of a tissue, to solve numerical modeling of casting solidification, to solve 2‐dimensional nonlinear obstacle problems, to solve the propagation of nonlinear water waves in numerical wave flume, to solve the shock‐induced 2‐dimensional coupled non‐Fickian diffusion‐elasticity, to simulate the 2‐dimensional sloshing phenomenon, to solve 2‐dimensional inverse Cauchy problems, to solve shallow water equations in 2 dimensions, to solve inverse Cauchy problems associated with 3‐dimensional Helmholtz‐type equations, or to solve inverse biharmonic boundary‐value problems …”

Section: Introductionmentioning

confidence: 99%

Adaptive strategies to improve the application of the generalized finite differences method in 2D and 3D

Ureña

Benito

Ureña

et al. 2017

Math Methods in App Sciences

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The generalized finite differences method allows the use of irregular clouds of nodes. The optimal values of the key parameters of the method vary depending on how the nodes in the cloud are distributed, and this can be complicated especially in 3D. Therefore, we establish 2 criteria to allow the automation of the selection process of the key parameters. These criteria depend on 2 discrete functions, one of them penalizes distances and the other one penalizes imbalances. In addition, we show how to generate irregular clouds of nodes more efficient than finer regular clouds of nodes. We propose an improved and more versatile h‐adaptive method that allows adding, moving, and removing nodes. To decide which nodes to act on, we use an indicator of the error a posteriori. This h‐adaptive method gives results more accurate than those ones presented for the generalized finite differences method so far and, in addition, with fewer nodes. In addition, this method can be used in time‐dependant problems to increase the temporal step or to avoid instabilities. As an example, we apply it in problems of seismic waves propagation.

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Application of the generalized finite difference method to solve the advection–diffusion equation

Cited by 86 publications

References 5 publications

Numerical study of detonation shock dynamics using generalized finite difference method

Numerical study of detonation shock dynamics using generalized finite difference method

Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

Adaptive strategies to improve the application of the generalized finite differences method in 2D and 3D

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