Abstract
In this paper, we propose a high order compact scheme for the unsteady two-dimensional reaction diffusion equations which is second order accurate in time and fourth order accurate in space. The scheme is specifically designed to tackle problems in pattern formation arising frequently in Mathematical Biology, modeled by the Gray–Scott model. By converting the reaction–diffusion equations into a pure diffusion equation, we obtain an unconditionally stable convergent implicit scheme. Though originally designed to capture the patterns generated by the reaction–diffusion equations, the scheme serves a dual purpose by efficiently capturing the incompressible viscous flows governed by the unsteady Navier–Stokes equations with equal ease. To validate the scheme, it is firstly applied to the famous lid-driven square cavity flow problem and then to two problems on pattern formation. Our computed results are compared with existing numerical ones and excellent match is obtained in all the cases.
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