An exponentially fitted difference scheme for the hyperbolic-hyperbolic singularly perturbed initial-boundary value problem (Q1192862)

The initial-boundary value problem of hyperbolic type with first derivative with respect to \(x\) \[ \varepsilon({\partial^ 2u\over \partial t^ 2}-{\partial^ 2u\over\partial x^ 2})+a(x,t){\partial u\over \partial t}+b(x,t){\partial u\over \partial x}+c(x,t)u=f(x,t), \] \((x,t)\in G=\{0<x<\ell,\;0<t\leq T\}\), \(u(x,0)=\varphi(x)\), \({\partial u\over \partial t}(x,0)=\psi(x)\), \(0\leq x\leq \ell\), \(u(0,t)=0\), \(u=(\ell,t)=0\), \(0\leq t\leq T\), is discussed. An energy estimate of its solution is given. The asymptotic solution is constructed under given compatibility conditions and uniform validity is proved in the sense of energy norm. An exponentially fitted difference scheme is developed and a discrete energy inequality is established. Uniform convergence of the discrete solution in the sense of discrete energy norm is proved finally.