Adaptive Mesh Generation and a Higher-Order Hybrid Approximation to Time-Dependent Singularly Perturbed Reaction–Diffusion Equations

Abstract This paper introduces a novel hybrid difference approximation for solving time-dependent reaction–diffusion problems with shifts and integral boundary conditions. The problem is singularly perturbed, and its solution exhibits multiscale behavior. The approach uses a backward difference discretization in time on a uniform mesh. A key feature is the generation of an adaptive moving mesh, partitioning the spatial domain to leverage the strengths of different discretization methods. It combines a cubic spline difference method within the boundary layer region and an exponential spline difference method in the outer layer region. The mesh is generated using the equidistribution principle. This approach enhances the accuracy of numerical solutions while maintaining computational efficiency. The paper also provides a thorough theoretical analysis and numerical results for four model problems. Numerical experiments confirm parameter uniform convergence and support the theoretical outcomes.