Efficient Adaptive Time-Stepping for Nonlinear Reaction-Diffusion Equations Using Crank-Nicolson Mixed FEM and Proper Orthogonal Decomposition

The paper presents an adaptive mixed Crank-Nicolson finite element approach (CNM-FEM) integrated with an appropriate orthogonal decomposition (POD) to efficiently solve the nonlinear reaction-diffusion problem. Because of their complexity and many unknowns, nonlinear reaction-diffusion equations pose major computing difficulties; they find applications in biology, chemistry, and physics among other domains. The proposed approach reduces this difficulty by dynamically changing the time step depending on error estimations over an adaptive time scale, hence improving computational efficiency while maintaining accuracy. The double-mesh technique, which solves nonlinear problems on a coarse mesh then refines them on a finer mesh, has improved the second-order accuracy and stability of the Crank-Nicolson method.By means of appropriate orthogonal decomposition (POD), system dimensionality is reduced, therefore enabling faster simulations without compromising solution quality and so reducing the computational load. Often found in real-world applications, Dirichlet and Neumann boundary conditions are addressed by this method. Along with more general numerical testing, benchmark problems include the Allen-Kahn equation and more challenging real-world scenarios highlight the accuracy, stability, and efficiency of the proposed approach. Comparisons with traditional fixed-time scaling techniques expose significant computing time savings especially in areas where the solution develops rapidly. The results confirm that an efficient and scalable framework for solving large-scale nonlinear interaction-diffusion problems with boundary conditions is provided by the adaptive hybrid Crank-Nicolson finite element approach with suitable orthogonal decomposition.