A uniformly convergent difference scheme for a semilinear singular perturbation problem
Consider a boundary value problem (1) \(F(y)=\epsilon y''-(a(x)y)'- b(x,y)=0\), \(0<x<1\), \(y(0)=A\), \(y(1)=B\), where \(a(x)\in C^ 2(0,1)\), \(b(x,y)\in C^ 2([0,1]\times R)\), \(R=(-\infty,\infty)\), \(0<\epsilon \ll 1\), \(b_ y(x,y)-| a(x)'| \geq \delta >0\) on [0,1]\(\times R\). Following a derivation principle of the Engquist-Osher scheme for equation (1) with \(a(x)y=f(y)\), a difference scheme for equation (1) is derived \((2)\quad F_ h(u)|_ i=(\epsilon /h^ 2)(\xi_{i+1}u_{i+1}-2\xi_ iu_ i+\xi_{i-1}u_{i-1})- (1/2h)(a_{i+1}u_{i+1}-a_{i-1}u_{i-1})-b(x_ i,u_ i)=0,\) \(i=0,...,N-1\), \(u_ 0=A\), \(u_ N=B\), where \(h=1/N\), \(\xi_ i=(a_ ih/2\epsilon)\coth (a_ ih/2\epsilon)\), \(a_ i=a(x_ i)\). Denoting by y(x) the solution of equation (1), the uniform convergence in \(\epsilon\) of the solution \(u=(u_ 0,...,u_ N)\) of equation (2) to y(x) is proved (Theorem 4). The results are illustrated by numerical examples.
semilinear singular perturbation problem
scientific article; zbMATH DE number 3901937