Structure-preserving numerical calculation of wave equation for a vector field

For the Proca equation, which is a wave equation for a vector field, we derive the canonical formulation including constraints from the Stueckelberg action and propose discrete equations with a structure-preserving scheme for conserving the constraints at the discrete level. Numerical simulations are performed using these discrete equations and other discrete equations with a standard scheme. We show the results obtained using the structure-preserving scheme and provide more accurate and stable numerical solutions.