A parameter-uniform implicit scheme for two-parameter singularly perturbed parabolic problems

Abstract

A parameter-uniform implicit approach for two-parameter singularly perturbed boundary valueproblems is constructed. On the solution derivatives, sharp limits are presented. The solution is additionallydivided into regular and singular components, limiting thederivatives of these components utilized in theconvergence analysis. In the temporal direction, the system of ordinary differential equations produced by theCrank-Nicolson scheme on a uniform mesh is further discretized in the spatial direction by employing a finitedifference technique on a selected Shishkin mesh. Through a rigorous analysis, we establish the theoreticalresults for two cases: Case I.ε1/ε22→0 asε2→0, and Case II.ε22/ε1→0 asε1→0, showing that thetechnique is convergent regardless of the magnitude of theε1, ε2parameters. The order of accuracy in Case Iand II are shown to beO((∆t)2+N−1(lnN)2) andO((∆t)2+N−2(lnN)2), respectively. Two examples arepresented to verify the theoretical results