Discretization error analysis for a radially symmetric harmonic map heat flow problem

In this paper we study the harmonic map heat flow problem for a radially symmetric case. The corresponding partial dfferential equation plays a key role in many analyses of harmonic map heat flow problems. We consider a basic discretization method for this problem, namely a second order finite difference discretization in space combined with a semi-implicit Euler method in time. The semi-implicit Euler method results in a linear problem in each time step. We restrict to the regime of smooth solutions of the continuous problem and present an error analysis of this discretization method. This results in optimal order discretization error bounds (apart from a logarithmic term). We also present discrete energy estimates that mimic the decrease of the energy of the continuous solution.