Compact schemes for anisotropic reaction–diffusion equations with adaptive time step

ELVIO HEIDENREICH, F.J. Gaspar, Ferrero de Loma-Osorio José María, J.F. Rodríguez

2010

International Journal for Numerical Methods in Engineering

Many problems in biology and engineering are governed by anisotropic reaction–diffusion equations with a very rapidly varying reaction term. These characteristics of the system imply the use of very fine meshes and small time steps in order to accurately capture the propagating wave avoiding the appearance of spurious oscillations in the wave front. This work develops a fourth-order compact scheme for anisotropic reaction–diffusion equations with stiff reactive terms. As mentioned, the scheme accounts for the anisotropy of the media and incorporates an adaptive time step for handling the stiff reactive term. The high-order scheme allows working with coarser meshes without compromising numerical accuracy rendering a more efficient numerical algorithm by reducing the total computation time and memory requirements. The order of convergence of the method has been demonstrated on an analytical solution with Neumann boundary conditions. The scheme has also been implemented for the solution of anisotropic electrophysiology problems. Anisotropic square samples of normal and ischemic cardiac tissue have been simulated by means of the monodomain model with the reactive term defined by Luo–Rudy II dynamics. The simulations proved the effectiveness of the method in handling anisotropic heterogeneous non-linear reaction–diffusion problems. Bidimensional tests also indicate that the fourth-order scheme requires meshes about 45% coarser than the standard second-order method in order to achieve the same accuracy of the results.