  We propose a new approach for solving systems of conservation laws that admit
a variational formulation of the time-discretized form, and encompasses the
p-system or the system of elastodynamics. The approach consists of using
constrained gradient descent for solving an implicit scheme with variational
formulation, while discontinuous Galerkin finite element methods is used for
the spatial discretization. The resulting optimization scheme performs well, it
has an advantage on how it handles oscillations near shocks, and a disadvantage
in computational cost, which can be partly alleviated by using techniques on
step selection from optimization methods.
