Free versus constrained evolution of the 2+1 equivariant wave map

Abstract

We compare the numerical solutions of the 2+1 equivariant wave map problem computed with the symplectic, constraint respecting Rattle algorithm and the well known fourth order Runge-Kutta method. We show the advantages of the Rattle algorithm for the constrained system compared to the free evolution with the Runge-Kutta method. We also present an expression, which represents the energy loss due to constraint violation. Taking this expression into account we can achieve energy conservation for the Runge-Kutta scheme, which is better than with the Rattle method. Using the symplectic scheme with constraint enforcement, we can reproduce previous calculations of the equivariant case without imposing the symmetry explicitly, thereby confirming that the critical behaviour is stable. © 2012 IOP Publishing Ltd.