Homework: Introduction to Multiscale Modeling - Singular Perturbation and Homogenization, Assignments of Health sciences

Download Homework: Introduction to Multiscale Modeling - Singular Perturbation and Homogenization and more Assignments Health sciences in PDF only on Docsity! Homework: Introduction to Multiscale Modeling 1) Consider the singular perturbation problem ! "# d 2 u dx 2 + du dx =1, 0 < x <1 u(0) = 0, u(1) = 0 (a) Determine the limit solution as ! " # 0, (" > 0) . (b) Determine an improved approximation by matched asymptotics for ! " = 0.1. Use the approximation from (a) for most of the interval and the solution of the homogeneous problem in an interval of length 0.2 close to the boundary with the boundary layer. Match these two approximations such that the final approximation is continuous and satisfies the boundary conditions. (c) Solve the original singular perturbation problem numerically by a finite difference approximation. Use ! " = 0.1 and the step size h=0.02. In the numerical approximation replace the derivatives in the original differential equation by divided differences, ! u(x j ) " u j , j = 0,..,J, hJ =1 d 2 u dx 2 # u j+1 $ 2u j + u j$1 h 2 , du dx # u j+1 $ u j$1 2h (d) Plot the approximations from (a), (b) and (c). 2) Consider the following two-point boundary problem with oscillatory coefficient, ! " d dx a(x /#) du dx $ % & ' ( ) =1, 0 < x <1 u(0) = 0, u(1) = 0 The oscillatory coefficient ! a(x /") is a one-periodic function given by, ! a(y) = 1, 0 " y < 0.5 0.5, 0.5 " y <1 # $ % (a) Determine the homogenized equation ! (" # 0) of the original two-point boundary problem. (b) Solve the homogenized equation analytically (c) Solve the original two-point boundary problem numerically with ! " = 0.1, h = 0.02 . Use the following finite difference formula,