A splitting based higher-order numerical scheme for 2D time-dependent singularly perturbed reaction-diffusion problems

The manuscript focuses on establishing an optimal accurate parameter-uniform scheme for singularly perturbed reaction-diffusion type problems in two dimensions. Along with the higher dimensions, these model problems have other complexities like the presence of multiple boundary layers because of the perturbation parameter. To handle these complexities, at first, the model problem is split into two lower-dimensional problems using the Peaceman–Rachford splitting algorithm based on the Crank–Nicolson scheme on a uniform mesh. Further, to be consistent with the second-order accuracy in the temporal direction, the spatial direction is addressed using the central difference scheme on some layer rectifying Shishkin-type meshes namely, the standard Shishkin mesh and the Bakhvalov-Shishkin mesh. The study extends with the application of the proposed scheme on a comparatively new Shishkin-type mesh, namely the Modified-Bakhvalov–Shishkin mesh. The use of the Bakhvalov–Shishkin-mesh and the Modified-Bakhvalov–Shishkin-mesh rectifies the logarithmic effect on the accuracy because of the standard Shishkin mesh. The scheme is made more computationally desirable by the use of robust Thomas algorithm. A global second-order accuracy is affirmed. All the theoretical claims are further corroborated in practice with some valid numerical tests and comparison results. In support of the claims, numerical experiments are also performed on model problems with larger time delays.