scholarly journals Computable exact bounds for linear outputs from stabilized solutions of the advection-diffusion-reaction equation

2012 ◽  
Vol 93 (5) ◽  
pp. 483-509 ◽  
Author(s):  
Núria Parés ◽  
Pedro Díez ◽  
Antonio Huerta
Keyword(s):  
Diffusion Reaction ◽  
Reaction Equation ◽  
Advection Diffusion ◽  
Exact Bounds

scholarly journals Numerical Simulation of Oil Spill in Ocean

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yong-Sik Cho ◽  
Tak-Kyeom Kim ◽  
Woochang Jeong ◽  
Taemin Ha
Keyword(s):  
Oil Spill ◽  
Shallow Water Equations ◽  
Dispersion Model ◽  
Coastal Environment ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Advection Diffusion ◽  
Coastal Damage ◽  
Spilled Oil ◽  
Marine Environmental

The spreading of oil in an open ocean may cause serious damage to a marine environmental system. Thus, an accurate prediction of oil spill is very important to minimize coastal damage due to unexpected oil spill accident. The movement of oil may be represented with a numerical model that solves an advection-diffusion-reaction equation with a proper numerical scheme. In this study, the spilled oil dispersion model has been established in consideration of tide and tidal currents simultaneously. The velocity components in the advection-diffusion-reaction equation are obtained from the shallow-water equations. The accuracy of the model is verified by applying it to a simple but significant problem. The results produced by the model agree with corresponding analytical solutions and field-observed data. The model is then applied to predict the spreading of an oil spill in a real coastal environment.

scholarly journals Performance of UPFD scheme under some different regimes of advection, diffusion and reaction

2017 ◽  
Vol 27 (7) ◽  
pp. 1412-1429 ◽  
Author(s):  
Appanah Rao Appadu
Keyword(s):  
Amplification Factor ◽  
Travelling Waves ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Step Size ◽  
Unconditionally Stable ◽  
Content Type ◽  
Advection Diffusion ◽  
Dispersion And Dissipation Errors ◽  
Relative Errors

Purpose An unconditionally positive definite finite difference scheme termed as UPFD has been derived to approximate a linear advection-diffusion-reaction equation which models exponential travelling waves and the coefficients of advection, diffusion and reactive terms have been chosen as one (Chen-Charpentier and Kojouharov, 2013). In this work, the author tests UPFD scheme under some other different regimes of advection, diffusion and reaction. The author considers the case when the coefficient of advection, diffusion and reaction are all equal to one and also cases under which advection or diffusion or reaction is more important. Some errors such as L1 error, dispersion, dissipation errors and relative errors are tabulated. Moreover, the author compares some spectral properties of the method under different regimes. The author obtains the variation of the following quantities with respect to the phase angle: modulus of exact amplification factor, modulus of amplification factor of the scheme and relative phase error. Design/methodology/approach Difficulties can arise in stability analysis. It is important to have a full understanding of whether the conditions obtained for stability are sufficient, necessary or necessary and sufficient. The advection-diffusion-reaction is quite similar to the advection-diffusion equation, it has an extra reaction term and therefore obtaining stability of numerical methods discretizing advection-diffusion-reaction equation is not easy as is the case with numerical methods discretizing advection-diffusion equations. To avoid difficulty involved with obtaining region of stability, the author shall consider unconditionally stable finite difference schemes discretizing advection-diffusion-reaction equations. Findings The UPFD scheme is unconditionally stable but not unconditionally consistent. The scheme was tested on an advection-diffusion-reaction equation which models exponential travelling waves, and the author computed various errors such as L1 error, dispersion and dissipation errors, relative errors under some different regimes of advection, diffusion and reaction. The scheme works best for very small values of k as k → 0 (for instance, k = 0.00025, 0.0005) and performs satisfactorily at other values of k such as 0.001 for two regimes; a = 1, D = 1, κ = 1 and a = 1, D = 1, κ = 5. When a = 5, D = 1, κ = 1, the scheme performs quite well at k = 0.00025 and satisfactorily at k = 0.0005 but is not efficient at larger values of k. For the diffusive case (a = 1, D = 5, κ = 1), the scheme does not perform well. In general, the author can conclude that the choice of k is very important, as it affects to a great extent the performance of the method. Originality/value The UPFD scheme is effective to solve advection-diffusion-reaction problems when advection or reactive regime is dominant and for the case, a = 1, D = 1, κ = 1, especially at low values of k. Moreover, the magnitude of the dispersion and dissipation errors using UPFD are of the same order for all the four regimes considered as seen from Tables 1 to 4. This indicates that if the author is to optimize the temporal step size at a given value of the spatial step size, the optimization function must consist of both the AFM and RPE. Some related work on optimization can be seen in Appadu (2013). Higher-order unconditionally stable schemes can be constructed for the regimes for which UPFD is not efficient enough for instance when advection and diffusion are dominant.

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