Polynomial Difference Schemes with Energy and Momentum Multipliers on Orthogonal Uniform Meshes: Stencil Restrictions and Monge--Ampère-Type Examples

Experience with invariant finite-difference schemes for one-dimensional hydro\-dynamic-type equations for a function $u(t,x)$ suggests that nonlinear polynomial schemes on orthogonal uniform meshes with the 9-point stencil cannot simultaneously preserve difference analogues of the energy and momentu