Ключевые слова: учет заполненности ячеек, разностная схема «кабаре», сеточные числа Пекле Работа выполнена при поддержке РФФИ (проект № 19-07-00623).
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.
Рассматривается развитие и применение метода учета заполненности прямоугольных ячеек материальной средой, в частности, жидкостью для повышения гладкости и точности конечноразностного решения задач гидродинамики со сложной формой граничной поверхности. Для исследования возможностей предлагаемого метода рассмотрены две задачи вычислительной гидродинамики - пространственно-двумерного течения вязкой жидкости между двумя соосными полуцилиндрами и пространственно-трехмерная задача волновой гидродинамики - распространения волны в прибрежной зоне и ее выхода на сушу. Для решения поставленных задач используются прямоугольные сетки, учитывающие заполненность ячеек. Аппроксимация задач по времени выполнена на основе схем расщепления по физическим процессам, а по пространственным переменным - на основе интегро-интерполяционного метода с учетом заполненности ячеек и без ее учета. Для оценки точности численного решения первой задачи в качестве эталона используется аналитическое решение, описывающее течение Куэтта-Тейлора. Моделирование производилось на последовательности сгущающихся расчетных сеток размерами: $11\times21$, $21\times41$, $41\times81$ и $81\times161$ узлов в случае применения метода и без его использования. В случае непосредственного использования прямоугольных сеток (ступенчатой аппроксимации границ) относительная погрешность расчетов достигает $70%$; при тех же условиях использование предлагаемого метода позволяет уменьшить погрешность до $6%$. Показано, что дробление прямоугольной сетки в $2$-$8$ раз по каждому из пространственных направлений не приводит к такому же повышению точности, которой обладают численные решения, полученные с учетом заполненности ячеек.
A 3D model of suspended matter transport in coastal marine systems is considered, which takes into account many factors, including the hydraulic size or the rate of particle deposition, the propagation of suspended matter, sedimentation, the intensity of distribution of suspended matter sources, etc. The difference operators of diffusion transport in the horizontal and vertical directions for this problem have significantly different characteristic spatiotemporal scales of processes, as well as spectra. With typical sampling, applied to shallow-water systems in the South of Russia (the Sea of Azov, the Tsimlyansk reservoir), the steps in horizontal directions are 200-1000 meters, the coefficients of turbulent exchange (turbulent diffusion) are (103-104) m2/sec; in the vertical direction - - - steps of 0.1 m-1 m, and the coefficients of microturbulent exchange in the vertical — (0.1-1) m2/sec. If we focus on the use of explicit locally twodimensional - - - locally one-dimensional splitting schemes, then the permissible values of the time step for a two-dimensional problem will be about 10-100 seconds, and for a one-dimensional problem in the vertical direction - - - 0.1 – 1 sec. This motivates us to construct an additive locally-two-dimensional-locallyonedimensional splitting scheme in geometric directions. The paper describes a parallel algorithm that uses both explicit and implicit schemes to approximate the two-dimensional diffusion-convection problem in horizontal directions and the one-dimensional diffusion-convection problem in the vertical direction. The two-dimensional implicit diffusion-convection problem in horizontal directions is numerically solved by the adaptive alternating-triangular method. The numerical implementation of the one-dimensional diffusion-convection problem in the vertical direction is carried out by a sequential run-through method for a series of independent one-dimensional three-point problems in the vertical direction on a given layer. To increase the efficiency of parallel calculations, the decomposition of the calculated spatial grid and all grid data in one or two spatial directions - in horizontal directions-is also performed. The obtained algorithms are compared taking into account the permissible values of time steps and the actual time spent on performing calculations and exchanging information on each time layer.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.