Free Versus Constrained Evolution of the 2+1 Equivariant Wave Map

TL;DR

This paper compares constrained and unconstrained numerical methods for the 2+1 equivariant Wave Map problem, highlighting the advantages of the symplectic Rattle algorithm in preserving constraints and energy conservation.

Contribution

It demonstrates the effectiveness of the Rattle algorithm for constrained wave map simulations and introduces an energy loss expression to improve energy conservation in Runge-Kutta methods.

Findings

Rattle algorithm better respects constraints in wave map simulations

Energy conservation improved by accounting for constraint violation

Symplectic scheme reproduces known critical behavior without explicit symmetry

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.