6.3 The constraints in the new systems: Numerical considerations

For numerical simulations, the role of the constraint equations is delicate: Since there are always numerical errors, although the vector field defined by Einstein’s equations in any of the above approaches is tangent to the constraint sub-manifold, we can only expect to be in the neighborhood of that sub-manifold. So, since one is effectively modifying the evolution equations outside the sub-manifold, that the vector field is tangent to it is not enough. For if that sub-manifold were unstable, it could very well be that a spurious numerical solutions could start growing during evolution and takes us completely away from it. This problem has been noticed by several people and has been considered in detail in [31]. There it is shown that, while in some evolution systems, the constraints themselves obey hyperbolic evolution equations, in others that is not the case so are presumably unstable.

It is not clear to me that the condition that the constraint system be well posed is the one needed for considering a system free of this problem. First because well posedness as such is not enough to guarantee the possibility of a numerical scheme: The system could be well posed but still depart exponentially from the constraint sub-manifold, thus making impossible any reliable calculation. So the non-principal part of the system must also be considered, and probably suitably modified in the neighborhood of the constraint sub-manifold. Second, since one is never solving, or simulating, the constraint evolution equations, that is, they play no role in the scheme, why should one consider them at all? I think the emphasis should be put on guaranteeing a numerical scheme without spurious solutions; because, as argued above, uniqueness should imply that the constraints are satisfied. Thus, what seems to be needed is a connection between well posedness, or rather no exponential departure from the constraint sub-manifold, and lack of spurious solutions on the numerical schema.

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