Optimization methods for solving systems of hydrodynamics nonlinear equations

There are two approaches to numerical solving steady state problems for the system of Navier–Stokes equations describing the motion of a viscous homogeneous incompressible fluid. One of them, that is most commonly used, is reduced to the solution of the non-stationary problem for the Navier–Stokes equations by an approximate method and obtaining steady solutions in the limit. Another, less popular, method is to construct a system of nonlinear algebraic equations (SNAE) using any approximation of the stationary problem solved; then, the resulting system is solved by iterative methods.
Each approach has its advantages and disadvantages. The main advantages of the second approach to the solution of stationary problems are: (1) in the process of convergence of the iterative process, the condition of noncompressibility is satisfied automatically; (2) one can use boundary conditions that are hard to implement in the usual way; (3) the method for solving stationary problems does not depend on the method of replacing the system of equations by a SNAE. Disadvantages of this approach are related to the fact that the iterative solutions of the SNAE for its convergence require restrictions on the nonlinear system operators and the initial approximation, which significantly limits the possibility of using this method for solving stationary problems.
The partial approximation method for solving difference problems approximating various stationary problems for the Navier–Stokes equations, based on the minimization of the residual norm minimization by specialized methods, was used earlier and proved to be effective. The possibility of solving difference problems approximating the stationary Navier–Stokes equations, multi-step iterative minimization possessing the properties of the CG method is investigated in this paper using the method of conjugate gradients and multi-step relaxation subgradient method (MSRSM).
The abovementioned optimization techniques have significant differences in the implementation and requirements to the accuracy of the one-dimensional descent. The conjugate gradient methods require a high precision search. The feature of relaxation subgradient methods, in particular, the MRSM, is the search of the direction of descent for the current gradient-coordinated minimum in a neighborhood the size of which is determined by the step of minimization. This last remark explains the effectiveness of methods of this class in the rough one-dimensional search.
Both the methods have successfully coped with the posed tasks within a reasonable time and high accuracy, which is confirmed by comparing the obtained results with known ones. These results allow us to expand the range of difference methods for solving problems approximating the stationary Navier–Stokes equations, which is particularly important under conditions of minimization of multiextremal functions, where it is often necessary to use several optimization methods.