Adjoint equations, integral conservation laws, and conservative difference schemes for nonlinear equations of mathematical physics

In this paper, we consider problems involving the construction of adjoint equations for nonlinear equations of mathematical physics. Hydrodynamical-type systems, in particular, dynamic equations for two-dimensional incompressible ideal fluid are taken as the main subject of investigation. It is shown that using adjoint equations, not only can we construct the known integrals of motion, but also obtain new integrals that are useful, in particular, for investigating the stability of solutions of the original equations. It is also shown that the nonuniqueness of the construction of adjoint equations for original nonlinear problems can be used to construct the finite-dimensional approximations of the original equations. These approximations have the necessary set of finite-dimensional analogues of integral conservation laws. The algorithm for constructing these schemes is given for a problem of two-dimensional ideal incompressible fluid dynamics.