A numerical study of mixed parabolic–gradient systems

This paper is concerned with the numerical solution of parabolic equations coupled with gradient equations. The gradient equations are ordinary differential equations whose solutions define positions of particles in the spatial domain of the parabolic equations. The vector field of the gradient equations is determined by gradients of solutions to the parabolic equations. Such mixed parabolic–gradient systems are for example, used in neurobiological studies of the formation of axonal connections in the nervous system. We discuss a numerical approach for solving parabolic–gradient systems on a grid. The basic ingredients are the fourth-order spatial finite differencing for the parabolic equations, piecewise cubic Hermite interpolation for approximating the gradient equations, and explicit time-stepping by means of a Runge–Kutta–Chebyshev method.