In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE. In particular, we consider a commonly used time dependent moving mesh method that aims to equidistribute the arclength of the solution under consideration. We assume that the discrete analogue of this equidistribution is strictly enforced, which allows us to reduce the effective dynamics to a scalar non-local problem with infinite range interactions. We show that this reduced problem is well-posed and obtain useful estimates on the resulting nonlinearities. In the sequel papers (Hupkes and Van Vleck in Travelling waves for adaptive grid discretizations of reaction diffusion systems II: linear theory; Travelling waves for adaptive grid discretizations of reaction diffusion systems III: nonlinear theory) we use these estimates to show that travelling waves persist under these adaptive spatial discretizations.

在本文中，我们为 Nagumo PDE 考虑具有自适应网格的空间离散化方案。特别是，我们考虑了一种常用的瞬态移动网格方法，该方法旨在均匀分布所考虑的解的弧长。我们假设这种等分布的离散模拟是严格执行的，这使我们能够将有效动力学减少为具有无限范围相互作用的标量非局部问题。我们表明，这个简化的问题是适定的，并获得对由此产生的非线性的有用估计。在后续论文中（Hupkes 和 Van Vleck 在用于反应扩散系统自适应网格离散化的行波 II：线性理论；用于反应扩散系统自适应网格离散化的行波 III：