Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem
Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem
复制标题
奇异摄动半线性反应扩散问题内层解的稳定近似
作者：
N. Kopteva;M. Stynes
考虑了一类二阶导数与一个小的正参数varepsilon^2}相乘的半线性反应扩散两点边值问题.它可以有多种解决方案。分析了具有内过渡层的解的数值计算。它表明，这种解决方案的精确计算是非常困难的。为了解决这个困难，我们提出了一个人工扩散稳定。对于标准的和稳定的有限差分方法在适当的Shishkin网格，我们证明存在性和调查计算的解决方案，通过构建离散的子和超级解决方案的准确性。收敛结果推导出依赖于${varepaly}$$和N的相对大小，其中N是网格间隔的数量。数值实验支持这些理论结果。讨论了用牛顿法计算离散解的实际问题。
A semilinear reaction–diffusion two-point boundary value problem, whose second-order derivative is multiplied by a small positive parameter $${varepsilon^2}$$ , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult. To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub- and super-solutions. Convergence results are deduced that depend on the relative sizes of $${varepsilon}$$ and N, where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton’s method to compute a discrete solution are discussed.