A fourth-order exponential time differencing scheme with real and distinct poles rational approximation for solving non-linear reaction-diffusion systems

A novel fourth-order, L-stable, exponential time differencing Runge–Kutta type scheme is developed to solve non-linear systems of reaction–diffusion equations, particularly with non-smooth data. The new scheme, ETDRK4RDP, is constructed by approximating the matrix exponentials in the ETDRK4 scheme with a fourth-order, L-acceptable, non-Padé rational function having real and distinct poles. The L-acceptable rational approximation ensures efficient damping of spurious oscillations arising from non-smooth initial and/or boundary conditions. The real and distinct poles of the rational function eliminate the presence of complex arithmetic in the solution of linear systems and thus make the scheme more attractive for parallelization in low-level languages like Fortran and C++. We verify empirically that the new ETDRK4RDP scheme is fourth-order accurate for several reaction diffusion systems with Dirichlet and Neumann boundary conditions and show it to be more efficient than competing exponential time differencing schemes when implemented in parallel, with up to five times speed-up in CPU time.