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 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.

arXiv comment16 pages, 8 figures. Formula for the scaling function on p. 13 corrected and two typos eliminated; otherwise agrees with the published paper

References in corpus (16)

• Global regularity of wave maps II. Small energy in two dimensions
• Numerical Investigation of Cosmological Singularities
• Renormalization and blow up for charge one equivariant critical wave maps
• Regularity of Wave-Maps in dimension 2+1
• Energy dispersed large data wave maps in 2+1 dimensions
• On "asymptotically flat" space-times with $G_2$-invariant Cauchy surfaces
• Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere
• New Algorithm for Mixmaster Dynamics
• Symplectic Integration of Post-Newtonian Equations of Motion with Spin
• Global Regularity of Wave Maps from R^{3+1} to H^2
• Singularity Formation in 2+1 Wave Maps
• On collapse of wave maps
• On the Formation of Singularities in the Critical O(3) Sigma-Model
• Free and constrained symplectic integrators for numerical general relativity
• Strongly hyperbolic Hamiltonian systems in numerical relativity: Formulation and symplectic integration
• On the applicability of constrained symplectic integrators in general relativity

Cited by in corpus (1)

• Multi-symplectic discretisation of wave map equations