Solving Hydrodynamic Problems Based on a Modified Upwind Leapfrog Scheme in Areas with Complex Geometry

Abstract

In recent years, the number of adverse and dangerous natural and anthropogenic phenomena has increased in coastal zones around the world. The development of mathematical modeling methods allows us to increase the accuracy of the study of hydrodynamic processes and the prediction of extreme events. This article discusses the application of the modified Upwind Leapfrog scheme to the numerical solution of hydrodynamics and convection–diffusion problems. To improve the accuracy of solving the tasks in the field of complex shapes, the method of filling cells is used. Numerical experiments have been carried out to simulate the flow of a viscous liquid and the transfer of substances using a linear combination of Upwind and Standard Leapfrog difference schemes. It is shown that the application of the methods proposed in the article allows us to reduce the approximation error in comparison with standard schemes in the case of large grid numbers of Péclet and to increase the smoothness of the solution accuracy at the boundary. The soil dumping and suspended matter propagation problems are solved using the developed schemes.

Research of the Solutions Proximity of Linearized and Nonlinear Problems of the Biogeochemical Process Dynamics in Coastal Systems

“…Combining these two models allows us to simulate the researched processes in a medium with a significant density gradient and a large difference in depths and considers the complex shape of the computational domain. The authors' team developed difference schemes that allow for the consideration of these features of coastal systems [24].…”

“…The results of the study of the oscillatory behavior of solutions of some classes of differential equations, which allow the modeling of many processes in the ocean, are presented in [23]. Numerical methods and difference schemes used to implement models of hydrodynamics and biogeochemistry are presented in [24][25][26].…”

“…Earlier, in the article [28], the authors' team presented a description of some aspects of the construction and numerical implementation of the model of biogeochemical cycles and biological kinetics of the multispecies population model. To this end, effective numerical methods and difference schemes have been developed that allow for the simulation of biogeochemical and hydrodynamic processes in a real computational domain of a complex shape [24]. However, some issues of the analytical study of the model remained in the shadows.…”

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

“…Changing the order of operations of differentiation and summation in (25), and calculating the derivative of a spatial variable:…”

“…The purpose of this work is to determine the range of the grid Péclet number values, in which the proposed scheme, which is a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and Standard Leapfrog scheme with 1/3 weight coefficient, obtained as a result of solving the problem of minimizing the approximation error, has better accuracy compared to classical schemes, including modifications of the Upwind Leapfrog scheme with limiters. Solutions of test problems of transport in areas of complex shape using the developed scheme are considered in the work [25]. These are problems such as the test problem of the fluid flow between two coaxial circular cylinders at different grid Péclet numbers, and the problem of suspended particles transport during soil dumping in areas with complex geometry.…”