反应系数同时依赖于时空坐标时扩散-反应方程的解析和数值结果

A. Askar, A. Nagy, I. F. Barna, Endre Kovács
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引用次数: 2

摘要

我们利用行波Ansatz得到了线性扩散反应方程的新的解析解。反应项同时是时间和空间的函数,首先是洛伦兹形式,其次是余弦行波形式。新解包含第一种情况下的Heun函数和第二种情况下的Mathieu函数,因此具有高度的非平凡性。我们利用这些解对标准显式和隐式方法进行了一些非常规的显式和稳定数值方法的测试,在标准显式和隐式方法中,后一种情况下的代数方程组是用预条件共轭梯度和GMRES求解器求解的。在此验证之后,我们还计算了受风冷却作用的二维表面的瞬态温度,该温度再次是空间和时间的函数。结果表明,显式稳定方法可以在较短的时间内达到隐式求解方法的数量级精度。
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Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates
We utilize the travelling-wave Ansatz to obtain novel analytical solutions to the linear diffusion–reaction equation. The reaction term is a function of time and space simultaneously, firstly in a Lorentzian form and secondly in a cosine travelling-wave form. The new solutions contain the Heun functions in the first case and the Mathieu functions for the second case, and therefore are highly nontrivial. We use these solutions to test some non-conventional explicit and stable numerical methods against the standard explicit and implicit methods, where in the latter case the algebraic equation system is solved by the preconditioned conjugate gradient and the GMRES solvers. After this verification, we also calculate the transient temperature of a 2D surface subjected to the cooling effect of the wind, which is a function of space and time again. We obtain that the explicit stable methods can reach the accuracy of the implicit solvers in orders of magnitude shorter time.
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