Singular Perturbations and Boundary Layers

Abstract

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Semi‐analytic time differencing methods for singularly perturbed initial value problems

“…To determine the fast part u fast , we follow the approach in References [10,11]: First, we insert (34) 2 into the difference equation of ( 1)-(35) 1 -(35) 2 , and write…”

“…To determine the fast part , we follow the approach in References [10, 11]: First, we insert (34) 2 into the difference equation of (1)–(35) 1 –(35) 2 , and write Now, collecting the terms of order and , we find the equations of and , …”

“…Note that our analysis and computations on (1) generalize significant amount the earlier work [10] as it includes the case when is singular and/or not diagonalizable. Using the techniques in singular perturbation analysis; see, for example, [11–18] as well as [19–28], we decompose the solution of (1) as the sum of a slow (non‐stiff) part and fast (stiff) part, and construct an explicit and smooth function, called, corrector which approximates the fast (stiff) part of . By embedding the explicitly built correctors in the conventional IF and ETD methods, and then apply the Runge–Kutta of order 2 and 4, we construct our semi‐analytic enriched IF and ETD schemes with Runge–Kutta methods.…”

Singularly perturbed reaction–diffusion problems on a k‐star graph

“…Subsequently, efforts have been made to understand the boundary or interior layers associated with various types of singularly perturbed differential equations, namely, viscous Burgers equations or Navier–Stokes equations. See, for example, Gie et al 8 for a recent survey and the references therein. The presence of boundary layers in fluid and gas dynamics can be attributed to the presence of the small perturbation parameter ϵ being multiplied to the highest order derivatives in the corresponding differential equations and, due to the limitation in obtaining analytic solutions, the numerical analysis of solutions has been actively pursued by research over the last few decades.…”

“…7 Subsequently, efforts have been made to understand the boundary or interior layers associated with various types of singularly perturbed differential equations, namely, viscous Burgers equations or Navier-Stokes equations. See, for example, Gie et al 8 for a recent survey and the references therein.…”