A Constraint-Preserving Finite Difference Method for the Damped Wave Map Equation to the Sphere

Abstract

We present and analyze a constraint-preserving finite difference method for approximating the damped wave map equation

$$\begin{aligned} \varepsilon u_{tt}+\alpha u_t - \varDelta u = \gamma u, \quad |u| = 1,\quad \text { in }(0,\infty )\times \varOmega , \end{aligned}$$

into the sphere. The numerical method preserves a discrete version of the energy balance associated with the equation and the unit length constraint of the solution at every grid point. We show that the approximations converge to a weak solution as the discretization parameters go to zero and present some numerical experiments investigating the limit $\varepsilon \rightarrow 0$ for $\alpha =1$.