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首页> 外文期刊>International journal of numerical methods for heat & fluid flow >Performance of UPFD scheme under some different regimes of advection, diffusion and reaction
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Performance of UPFD scheme under some different regimes of advection, diffusion and reaction

机译：UPFD方案在对流，扩散和反应某些不同状态下的性能

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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 L_1error, 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 L_1error, 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, k = 1 and a = 1, D = 1, k = 5. When a = 5, D = 1, k = 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, k = 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, k = 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.
机译：目的-推导了称为UPFD的无条件正定有限差分方案，以近似模拟对数行波的线性对流-扩散-反应方程，并且选择了对流，扩散和反应项的系数（Chen-Charpentier和Kojouharov，2013年）。在这项工作中，作者在对流，扩散和反应的其他一些不同机制下测试了UPFD方案。作者考虑了平流，扩散和反应系数都等于1的情况，以及平流，扩散或反应系数更重要的情况。表中列出了一些误差，例如L_1误差，色散，耗散误差和相对误差。此外，作者比较了在不同方案下该方法的一些光谱特性。作者获得了以下有关相位角的量的变化：精确放大系数的模数，方案放大系数的模数和相对相位误差。设计/方法/方法-稳定性分析可能会遇到困难。充分了解获得的稳定性条件是否充分，必要或必要和充分是很重要的。对流扩散反应与对流扩散方程非常相似，它具有一个额外的反应项，因此要获得离散化对流扩散反应方程的数值方法的稳定性并不容易，就像离散化对流扩散的数值方法一样。方程。为避免在获得稳定区域时遇到困难，作者应考虑无条件稳定的有限差分格式，将对流扩散反应方程式离散化。结果-UPFD方案无条件稳定，但并非无条件一致。该方案在对流-扩散-反应方程中进行了测试，该方程模拟了指数行波，作者计算了各种误差，例如L_1误差，色散和耗散误差，在不同对流，扩散和反应机制下的相对误差。该方案对于k的极小值k！0（例如，k = 0.00025、0.0005）最有效，并且在两种情况下，在k的其他值（例如0.001）下也能令人满意地执行； a = 1，D = 1，k = 1，a = 1，D = 1，k =5。当a = 5，D = 1，k = 1时，该方案在k = 0.00025时表现良好，在k处令人满意= 0.0005，但在较大的k值时效率不高。对于扩散情况（a = 1，D = 5，k = 1），该方案不能很好地执行。通常，作者可以得出结论，k的选择非常重要，因为它在很大程度上影响该方法的性能。原创性/价值-当对流或反应性占主导地位时，对于a = 1，D = 1，k = 1的情况，尤其是在k值较低的情况下，UPFD方案可有效解决对流扩散反应问题。此外，从表1至表4可以看出，对于所有四种情况，使用UPFD的色散和耗散误差的大小都处于同一数量级。这表明，如果作者要在给定的条件下优化时间步长，在空间步长上，优化功能必须同时包含AFM和RPE。有关优化的一些相关工作可以在Appadu（2013）中看到。对于UPFD效率不高的情况（例如在对流和扩散占主导地位的情况下），可以构造更高阶的无条件稳定方案。

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