The stability and convergence of the finite analytic method for the numerical solution of convective diffusion equation

A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.

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Solutions of Navier-Stokes Equation with Coriolis Force

Advances in Mathematical Physics ◽

10.1155/2017/7042686 ◽

2017 ◽

Vol 2017 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Sunggeun Lee ◽

Shin-Kun Ryi ◽

Hankwon Lim

Keyword(s):

Diffusion Equation ◽

Stokes Equation ◽

Coriolis Force ◽

Potential Flow ◽

Convective Diffusion ◽

Navier Stokes ◽

Coriolis Effect ◽

Two Dimensional ◽

Navier Stokes Equation ◽

Convective Diffusion Equation

We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, Ω. In both steady state and two-dimensional flow, the vorticity vector ω gets shifted by the amount of -2Ω. Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow v→=∇→ϕ, the equation satisfied by ϕ is independent of Ω. The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration c and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of c.

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Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

Mathematics ◽

10.3390/math8020187 ◽

2020 ◽

Vol 8 (2) ◽

pp. 187

Author(s):

Yaxin Hou ◽

Cao Wen ◽

Hong Li ◽

Yang Liu ◽

Zhichao Fang ◽

...

Keyword(s):

Finite Element ◽

Diffusion Equation ◽

Mixed Finite Element ◽

Second Order ◽

Stability And Convergence ◽

Numerical Examples ◽

The Stability ◽

High Order Time ◽

Distributed Order ◽

Convergence Of The Scheme

In this article, some high-order time discrete schemes with an H 1 -Galerkin mixed finite element (MFE) method are studied to numerically solve a nonlinear distributed-order sub-diffusion model. Among the considered techniques, the interpolation approximation combined with second-order σ schemes in time is used to approximate the distributed order derivative. The stability and convergence of the scheme are discussed. Some numerical examples are provided to indicate the feasibility and efficiency of our schemes.

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