An Implicit Scheme for Solving the Convection–Diffusion–Reaction Equation in Two Dimensions

Sheu, Tony W. H.; Wang, S. K.; Lin, R. K.

doi:10.1006/jcph.2000.6588

Cited by 47 publications

(13 citation statements)

References 17 publications

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“…Note that use of m given in Equation (22) can avoid the complex variable problem encountered in our previous article [3]. The present scheme involves calculating x (≡ Fh) and xx (≡ Gh 2 ) shown on the right-hand sides of Equations (10), (14) and (17).…”

Section: One-step Second-order Temporal Schemementioning

confidence: 98%

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

Lin¹,

Sheu²

2006

Numerical Methods in Fluids

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SUMMARYIn this paper we present a class of semi-discretization ÿnite di erence schemes for solving the transient convection-di usion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection-di usion (CD) equation to the inhomogeneous steady convectiondi usion-reaction (CDR) equation after using di erent time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed timestepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model di erential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection-di usion-reaction di erential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions.

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Section: One-step Second-order Temporal Schemementioning

confidence: 98%

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

Lin¹,

Sheu²

2006

Numerical Methods in Fluids

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“…(3.6) be u(x, t) = e Iθx e αt , see [20]. Then u t = αe αt e Iθx , u x = Iθe Iθx e αt , u xx = (Iθ) 2 e Iθx e αt and u xxx = (Iθ) 3 e Iθx e αt .…”

Section: Numerical Dispersionmentioning

confidence: 99%

Some optimised schemes for 1D Korteweg-de-Vries equation

Appadu

Chapwanya

Jejeniwa

2017

PCFD

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Two new explicit finite difference schemes for the solution of the one dimensional Korteweg-de-Vries equation are proposed. This equation describes the character of a wave generated by an incompressible fluid. We analyse the spectral properties of our schemes against two existing schemes proposed by Zabusky and Kruskal (Phys. Rev. Lett. 15(6):240-243,1965) and Wang et al. (Chinese Phys. Lett. 25(7):2335-2338, 2008). An optimization technique based on minimisation of the dispersion error is implemented to compute the optimal value of the spatial step size at a given value of the temporal step size and this is validated by some numerical experiments. The performance of the four methods are compared in regard to dispersive and dissipative errors and their ability to conserve mass, momentum and energy by using two numerical experiments which involve solitons.

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“…Within the mixed formulation, where the divergence-free constraint equation (1) is solved together with the momentum equations (13)- (14), inclusion of the constraint equation (1) will increase the matrix condition number and matrix size. To overcome this computational difficulty, we adopt the conventional segregated approach by reformulating the mass conservation equation in terms of the pressure by virtue of (q=qx) (13) þ (q=qy) (14) and then employing Eq.…”

Section: Electro-osmotically Driven Microchannel Flow 505mentioning

confidence: 99%

“…To overcome this computational difficulty, we adopt the conventional segregated approach by reformulating the mass conservation equation in terms of the pressure by virtue of (q=qx) (13) þ (q=qy) (14) and then employing Eq. (1).…”

Section: Electro-osmotically Driven Microchannel Flow 505mentioning

confidence: 99%

Numerical Study of an Electro-Osmotically Driven Microchannel Flow with Joule Heating Effect

Sheu

Kuo

2009

Numerical Heat Transfer, Part B: Fundamentals

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A convection-diffusion reaction scheme is applied to solve the transient transport equations for the prediction of steady electro-osmotic microchannel flow behavior. The governing equations for the total electric field include the Laplace equation for the effective electrical potential and the Poisson-Boltzmann equation for the electrical potential established in the electric double layer. The transport equations governing the hydrodynamic field variables comprise mass conservation equation for the electrolyte and equations of motion for the incompressible charged fluid flow subject to an electro-osmotic body force. The main aim of the study is to elucidate the effect of Joule heating, which can affect the electrohydrodynamic behavior. Investigation into the region near the negatively charged channel wall is made through the simulated velocity boundary layer, diffuse layer, and electric double layer.

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An Implicit Scheme for Solving the Convection–Diffusion–Reaction Equation in Two Dimensions

Cited by 47 publications

References 17 publications

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

Some optimised schemes for 1D Korteweg-de-Vries equation

Numerical Study of an Electro-Osmotically Driven Microchannel Flow with Joule Heating Effect

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