We present an overview of recent developments in the numerical solution of
Horndeski gravity theories, which are the class of all scalar-tensor theories
of gravity that have second order equations of motion. We review several
methods that have been used to establish well-posed initial value problems for
these theories, and discuss well-posed formulations of the constraint
equations. We also discuss global aspects of exact, strongly coupled solutions
to some of Horndeski gravity theories: the formation of shocks, the loss of
hyperbolicity, and the formation of naked curvature singularities. Finally we
discuss numerical solutions to binary black hole and neutron star systems for
several Horndeski theories.



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