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首页> 外文期刊>Journal of dynamics and differential equations >Travelling Waves for Complete Discretizations of Reaction Diffusion Systems
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Travelling Waves for Complete Discretizations of Reaction Diffusion Systems

机译：行波用于反应扩散系统的完全离散化

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In this paper we consider the impact that full spatial-temporal discretizations of reaction-diffusion systems have on the existence and uniqueness of travelling waves. In particular, we consider a standard second-difference spatial discretization of the Laplacian together with the six numerically stable backward differentiation formula methods for the temporal discretization. For small temporal time-steps and a fixed spatial grid-size, we establish some useful Fredholm properties for the operator that arises after linearizing the system around a travelling wave. In particular, we perform a singular perturbation argument to lift these properties from the natural limiting operator. This limiting operator is associated to a lattice differential equation, where space has been discretized but time remains continuous. For the backward-Euler temporal discretization, we also obtain travelling waves for arbitrary time-steps. In addition, we show that in the anti-continuum limit, in which the temporal time-step and the spatial grid-size are both very large, wave speeds are no longer unique. This is in contrast to the situation for the original continuous system and its spatial semi-discretization. This non-uniqueness is also explored numerically and discussed extensively away from the anti-continuum limit.
机译：在本文中，我们考虑了反应扩散系统的完整时空离散化对行波的存在和唯一性的影响。特别是，我们考虑了拉普拉斯算子的标准二次差分空间离散化以及用于时间离散化的六种数值稳定的向后差分公式方法。对于较小的时间步长和固定的空间网格大小，我们为操作员建立了一些有用的Fredholm属性，这些属性在围绕行波线性化系统之后产生。特别地，我们执行奇异摄动参数以从自然极限算符中提升这些特性。该限制算子与一个晶格微分方程相关联，在该方程中空间已经离散化，但时间保持连续。对于后向欧拉时间离散，我们还获得了任意时间步长的行波。此外，我们表明在反连续极限中，时间步长和空间网格大小都非常大，波速不再是唯一的。这与原始连续系统及其空间半离散化的情况相反。还从数字上探索了这种非唯一性，并在远离反连续性极限的范围内进行了广泛讨论。

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