New Hermite collocation approach with shocks wave capturing for solving non-linear coupled Burgers-type model at high Reynolds number

This paper aims to study the shock wave type nonlinear coupled advection–diffusion models using the cubic Hermite-based collocation method. This approach combines orthogonal collocation within the finite element framework with Hermite basis functions of degree three to ensure the continuity of a dependent variable and its first derivative throughout the solution domain. The cubic Hermite collocation method (CHCM) is utilized for spatial direction, while time discretization is implemented through the Crank–Nicolson scheme. The stability and convergence of the proposed algorithm are analyzed, revealing that it is unconditionally stable and possesses an order of \(\mathcal {O} \left( h^{2}+(\Delta t)^2 \right)\) $\mathcal{O}\left(h^2,+,{(\mathrm{\Delta }t)}^2\right)$ . Several experiments are conducted to showcase the accuracy of CHCM by comparing the results with exact solutions or those from existing literature. In the experiments, the viscosity parameter is varied up to \(10^{-4}\) ${10}^{-4}$ , and from the figures, it is evident that the CHCM efficiently captured shock waves. The obtained results validate the theoretical ones. It has been determined that the proposed technique is straightforward to implement and yields superior results as compared to existing methods.