A uniform convergent approach for system of singularly perturbed convection diffusion problem with interior turning point

This article presents an efficient and higher-order accurate numerical technique for solving a system of singularly perturbed convection–diffusion problems with an interior turning point. The approach combines the θ-method for time discretization with an exponential B-spline (EBS) collocation scheme on a Shishkin mesh for spatial approximation. Unlike standard finite difference and classical spline-based methods, the present approach provides parameter-uniform convergence, maintaining accuracy even for very small perturbation parameters where many existing methods suffer from layer-induced errors. Moreover, the integration of the Shishkin mesh with EBS collocation enables accurate resolution of both boundary layers without spurious oscillations, which is often challenging for conventional discretization techniques. The scheme is unconditionally stable for 1/2⩽θ⩽1 and achieves first-order convergence in time for 1/2<θ⩽1, and second-order accuracy for the Crank–Nicolson case θ=1/2, along with nearly second-order parameter-uniform convergence in space. Numerical experiments show that the maximum error decreases from 10−1 to 10−3 in space and from 10−2 to 10−5 in time as the mesh is refined, uniformly for perturbation parameters ranging from 2−8 to 2−16. The method maintains stable and accurate solutions with moderate computational cost, with CPU times remaining within practical limits even for fine discretizations. These results indicate that the proposed scheme offers improved accuracy, robustness, and is well suited for practical applications involving sharp gradients, such as heat transfer, fluid flow, and transport in porous media.