In order to improve the approximation of spatial derivatives without meshes, a set of meshfree numericalschemes for derivative terms is developed, which is compatible with the coordinates of Cartesian, cylindrical, andspherical. Based on the comparisons between numerical and theoretical solutions, errors and convergences areassessed by a posteriori method, which shows that the approximations for functions and derivatives are of thesecond accuracy order, and the scale of the support domain has some influences on numerical errors but not onaccuracy orders. With a discrete scale h=0. 01, the relative errors of the numerical simulation for the selectedfunctions and their derivatives are within 0. 65%.

In the present paper, a basic proof method is provided for representing the Verification, validation and evaluation of expert systems. The result provides an overview of the basic method for formal proof such as: partition larger systems into small systems prove correctness on small systems by non-recursive means, prove that the correctness of all subsystems implies the correctness of the entire system.

This paper presents the results of the full-scale Verification and validation of the mathematical model and numerical algorithm for the network computation of a single-phase flow in a highly branched pipeline chain. The essential difference of this work from others is that highly branched hydraulic networks with homogeneous and non-uniform permeability, containing more than 70 thousand branches, are considered. Such branched networks are important in many applications. Therefore, the development of algorithms for calculating flows in such networks is very important. The network model is based on hydraulic theory, and the numerical algorithm relies on the network analogue of the well-known control volume method. At the same time, obtaining reliable experimental data for testing models for calculating very branched hydraulic networks is very difficult. In this work, microfluidic technologies are used to solve this problem. Data of laboratory experiments, obtained using microfluidic models of branched networks with homogeneous and heterogeneous permeability, containing several tens of thousands of branches, as well as CFD simulation results in full 3D formulation employing the fine computational grids were used to validate the model. The Reynolds number ranged from 0.81 to 13. Conducted validation has shown a good qualitative and quantitative concordance of the results of network and hydrodynamic simulation, as well as the data of the microfluidic experiments. The error in determining the total pressure drop in the branched hydraulic network with heterogeneous permeability, containing 37,855 nodes and 74,900 branches, did not exceed 5%. It has been demonstrated that the speed of solving a single-phase flow problem in a highly branched chain using network simulation techniques is 60 times more of magnitude higher as compared to CFD simulation at virtually the same accuracy.