A difference method for solving the convection-diffusion equation with a nonclassical boundary condition in a multidimensional domain

The paper studies a multidimensional convection-diffusion equation with variable coefficients and a nonclassical boundary condition. Two cases are considered: in the first case, the first boundary condition contains the integral of the unknown function with respect to the integration variable $x_ \alpha$, and in the second case, the integral of the unknown function with respect to the integration variable $\tau$, denoting the memory effect. Similar problems arise when studying the transport of impurities along the riverbed. For an approximate solution of the problem posed, a locally one-dimensional difference scheme by A. A. Samarskii with order of approximation $O(h^2+\tau)$. In view of the fact that the equation contains the first derivative of the unknown function with respect to the spatial variable $x_\alpha$, the well-known method proposed by A. A. Samarskii in constructing a monotonic scheme of the second order of accuracy in $h_\alpha$ for a general parabolic type equation containing one-sided derivatives taking into account the sign of $r_\alpha(x,t)$. To increase the boundary conditions of the third kind to the second order of accuracy in $h_\alpha$, we used the equation, on the assumption that it is also valid at the boundaries. The study of the uniqueness and stability of the solution was carried out using the method of energy inequalities. A priori estimates are obtained for the solution of the difference problem in the $L_2$-norm, which implies the uniqueness of the solution, the continuous and uniform dependence of the solution of the difference problem on the input data, and the convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem in the $L_2$-norm with speed equal to the order of approximation of the difference scheme. For a two-dimensional problem, a numerical solution algorithm is constructed.