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Error Driven Node Placement as Applied to One Dimensional Shallow Water Equations
Abstract
This paper investigates the technique of localised mesh refinement to solve the Shallow Water Equations (SWE), by adding nodes only where needed. The discretization process linearizes the nonlinear equations for solving as a linear system. The nonlinear error values at specific nodes are used to indicate
which node will have additional nodes added either either side. The process of adding nodes is repeated until the Nonlinear error value is below a given threshold, or the predefined maximum number of nodes for that given time step has been reached. This process is restarted again at each time step, allowing the optimization process to efficiently allocate nodes based only on error, avoiding global increases in node numbers
across the solution set.
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This article reports the behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations. The one-and-a-half- dimensional shallow water equations are the onedimensional shallow water equations with a passive tracer or transverse velocity. The studied behaviour is with respect to the choice of numerical fluxes to evolve the mass, momentum, tracer-mass (transverse momentum), and entropy. When solving the one-and-a-half-dimensional shallow water equations using a finite volume method, we recommend the use of a double sided stencil flux for the mass and momentum, and in addition, a single sided stencil (upwind) flux for the tracer-mass. Having this recommended combination of fluxes, we use a double sided stencil entropy flux to compute the numerical entropy production, but this flux generates positive overshoots of the numerical entropy production. Positive overshoots of the numerical entropy production are avoided by use of a modified entropy flux, which satisfies a discrete numerical entropy inequality.
Mesh generation is one of key issues in Computational Fluid Dynamics. This paper presents an adaptive two-dimensional mesh refinement method based on the law of mass conservation. The method can be applied to a governing system that includes the law of mass conservation (continuity equation) for incompressible or compressible steady flows. Users can choose how many refinements they want to perform on the original mesh. The more the number of refinements, the less the error of calculations is. The refined meshes can identify the accurate locations of asymptotes and singular points and draw accurate closed streamlines if the number of refinements is big enough. We show two examples that demonstrate the claims.
Mesh generation is one of key issues in Computational Fluid Dynamics. This paper presents an adaptive three-dimensional mesh refinement method based on the law of mass conservation. The method can be applied to a governing system that includes the law of mass conservation (continuity equation) for incompressible or compressible steady flows. Users can choose how many refinements they want to perform on the initial mesh. The more the number of refinements, the less the error of calculations is. The refined meshes can identify the accurate locations of asymptotes, the points at which one, two, or all components of velocity fields are equal to zero, and draw accurate closed streamlines if the number of refinements is big enough and the initial mesh is fine enough. We show three examples that demonstrate the claims.
In this study a finite difference method for solving initial boundary value problem for the one-dimensional nonlinear system of differential equations describing shallow water flows in a class of discontinuous functions is suggested. In order to find the numerical solution of the problem, known as main problem, a special auxiliary problem is introduced. The degree of smoothness of the solution of the auxiliary problem is higher than the smoothness of the solution of the main problem. Moreover, the suggested auxiliary problem makes it possible to write out the effective and higher order numerical algorithms. Some results of numerical experiments are demonstrated.
Contenido: Conceptos básicos de corrientes de fluidos; Introducción a los métodos numéricos; Métodos de diferencia finita; Métodos de volumen finito; Solución de sistemas de ecuaciones lineales; Método de problemas inestables; Solución de la ecuación de Navier-Stokes; Geometrías complejas; Flujos turbulentos; Flujo comprensible; Eficiencia y mejora de la exactitud; Cuestiones especiales; Apéndices.
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