The finite difference streamline diffusion methods for Sobolev equations with convection-dominated term

Sun, Tongjun; Yang, Dinghui

doi:10.1016/s0096-3003(00)00135-1

Cited by 36 publications

(21 citation statements)

References 8 publications

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“…Brill [8] and Showalter [48] established the existence of solutions of semilinear evolution equations of Sobolev type in Banach space, while the global solvability and blow-up of equations of Sobolev type were considered in [2]. Numerically, a lot of simulation methods have been developed for pseudo-parabolic equations [62,46], but most of the existing works are based on the classical FE methods [47,30], FD schemes [50], or FV element methods [60] as discretization tools.…”

Section: Introductionmentioning

confidence: 99%

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Garmanjani

Cavoretto

Esmaeilbeigi

2018

Computers & Mathematics with Applications

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Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations.

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Section: Introductionmentioning

confidence: 99%

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Garmanjani

Cavoretto

Esmaeilbeigi

2018

Computers & Mathematics with Applications

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“…Various finite element and finite volume schemes have been constructed to treat such problems (see [3][4][5][6]) and global superconvergence were also studied for one and two-dimensional problems in [4,7]. We can see other numerical methods such as spectral and finite difference approximations of this type of equations in [1,8,9].…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of spectral radial point interpolation method locally applied on two‐dimensional pseudoparabolic equation

Shivanian

Aslefallah

2016

Numerical Methods Partial

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In this article, we study a spectral meshless radial point interpolation of pseudoparabolic equations in two spatial dimensions. Shape functions, which are constructed through point interpolation method using the radial basis functions, help us to treat problem locally with the aim of high‐order convergence rate. The time derivatives are approximated by the finite difference time‐stepping method. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical results are presented to illustrate the theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 724–741, 2017

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“…[3, 4, 5, 6, 7]. Because the convection term d ( x ) ⋅∇ u does exist in many practical applications, the numerical approaches for Sobolev equations with convection term have drawn much attention and have been given among others by Nakao [8] using finite element method for some nonlinear Sobolev equations in one space dimension, Gu [9] using the characteristic finite element method which is based on the approximation of the material derivative term, Sun and Yang [10] who proposed two finite difference streamline diffusion schemes by using the streamline diffusion finite element methods in space and finite difference schemes in time, Gao and Rui [11] using the split least‐squares characteristic mixed finite element method, Sun and Ma [12] who proposed a space‐time discontinuous Galerkin finite element scheme, and Zhang et al [13] using continuous interior penalty finite element method.…”

Section: Introductionmentioning

confidence: 99%

An expanded mixed covolume method for sobolev equation with convection term on triangular grids

Fang

2012

Numerical Methods Partial

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The expanded mixed covolume method for the two-dimensional Sobolev equation with convection term is developed and studied. This method uses the lowest-order Raviart-Thomas mixed finite element space as the trial function space. By introducing a transfer operator γ h which maps the trial function space into the test function space and combining expanded mixed finite element with mixed covolume method, the continuousin-time, discrete-in-time expanded mixed covolume schemes are constructed, and optimal error estimates for these schemes are obtained. Numerical results are given to examine the validity and effectiveness of the proposed schemes.

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The finite difference streamline diffusion methods for Sobolev equations with convection-dominated term

Cited by 36 publications

References 8 publications

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Stability and convergence of spectral radial point interpolation method locally applied on two‐dimensional pseudoparabolic equation

An expanded mixed covolume method for sobolev equation with convection term on triangular grids

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