Uniform difference scheme for a singularly perturbed linear 2nd order hyperbolic problem with zeroth order reduced equation

A uniformly convergent finite difference method for a singularly perturbed initial value problem

“…The difference scheme for this type of problems has been studied in [ 4 ]. In the same paper for linear hyperbolic equation, which reduces for e = 0 to zero-order equation was proved only O( r TM) convergence for difference solution, when the second initial condition doesn't contain s.…”

“…The difference scheme as well its method of construction differs from those in [ 4 ] and the scheme has uniform O(r) accuracy. The difference scheme is constructed by the method of integral identities with the use exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form [7 ] -[ 9].…”

“…Several uniform difference methods for problems with initial layers have been proposed in [4]- [7]. To our knowledge, the only reference to a uniformly convergent scheme for the second order differential equations, which reduce for = 0 to zero-order equations is [5], but this scheme correspond to the particular case when the second initial condition is regular and furthermore only is O(τ 1/4 ) accurate uniformly in , where τ is the mesh size.…”