Novel high-order compact approach for dynamics of spiral waves in excitable media

Abstract

• Comprehensive spectral density analysis of the spiral tip to investigate the dynamics of the spiral wave. • Time and space grid independent computation. • Modified differential equation analysis of the explicit scheme for reaction-diffusion equations. • Numerical study on the stability of rotating spiral wave solutions. • Effect of an obstacle to the periodic rotating spiral waves. The current study envisages to investigate the dynamics of spiral wave in and without the presence of obstacles in excitable media through a comprehensive spectral density analysis. While most of the previous studies on spiral wave dynamics are seen to have been obtained by exploiting the state variables using lower order explicit schemes to discretize the governing equations, we accomplish the same by exploiting the spiral tip path with data obtained from a recently developed higher order compact finite difference scheme. This scheme which is implicit in nature and unconditionally stable is seen to efficiently resolve the spiral wave patterns. On the other hand, owing to implicit dispersion, diffusion and reaction terms inherently present in the explicit data, the spiral tip resulting from it was seen to behave differently for different time steps. These issues are illustrated through a modified differential equation analysis of the explicit scheme. For the range of the parameters considered, we further observed that the role of the obstacle was only to change the trajectory paths, as they were seen to settle into a periodic motion eventually. In the process, we also establish the grid independence and the rate of convergence of the simulated data, apart from exploring the stability of the computed rotating spiral wave solutions.