A new angular momentum method for computing wave maps into spheres (Q2927846)

The purpose of the paper is to develop a new numerical method for computing the solution of a constrained wave equation \(d_{tt} - \Delta d = \gamma d\), \(|d|=1\), defined on a unit box or the torus \(\mathbb{T}^n\), \(n=2,3\). Here \(\gamma\) is a Lagrange multiplier enforcing the constraint \(|d|=1\). This problem is highly nonlinear and their solutions are not smooth. By introducing the angular momentum, the previous wave equation is rewritten as a first-order system which does not require the Lagrange multiplier \(\gamma\). Then, a particular discretization using forward differences in time is proposed that preserves both the energy and the constraint \(|d|=1\). The discretization in space uses a standard central difference approximation of the Laplacian on a regular grid.NEWLINENEWLINEThe resulting numerical scheme is nonlinear and implicit, so that it requires a fixed point solver for its actual implementation. It is shown how this solver can be constructed with the required properties using only \(N \log N\) operations, \(N\) being the number of degrees of freedom of \(d\). This angular momentum method converges to a weak solution of the continuous problem as discretization parameters go to zero.