Computational aspects of constructing field gradients in the numerical implementation of variational algorithms for identifying the parameters of the transfer model

When solving environmental problems, methods of numerical modeling of dynamic processes in reservoirs, modeling the spread of various impurities and pollutants occupy an important place. A special place in this case is occupied by methods of assimilation of measurement data in numerical models. One of the approaches to solving such problems is the variational approach. Such algorithms are based on solving conjugate problems and problems in variations in the identification of numerical simulation parameters. The search for such parameters is carried out by minimizing the prediction quality functional, which characterizes the deviations of model estimates of the state from the measurement data. When implementing the variational algorithm, the gradient components in the parameter space are determined. If the identification of the initial field requires the presence of the solution of the conjugate problem itself, then when determining the coefficients of turbulent diffusion, it is necessary to calculate the spatial derivatives of the concentration field itself, as well as the solutions of the conjugate problem. In this paper, we consider the difference discretization of the passive impurity transfer equation and the consistent approximation of the gradients of the fields of solutions to the main and associated problems in the implementation of a variational algorithm for assimilation of measurement data and identification of model parameters. A wide parametric family of difference schemes for integrating the used transfer model with certain properties at different parameter values is presented. The corresponding approximations of the gradients of the fields necessary for the implementation of the variational identification algorithm are written out.