Numerical study of slightly viscous flow

Chorin, Alexandre J.

doi:10.1017/s0022112073002016

Cited by 1,294 publications

(823 citation statements)

References 7 publications

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“…Nevertheless, the PVM is quite unstable in general. A vortex blob was introduced by Chorin in 1973 [7] to replace the point vortex, known as vortex blob method, and was successfully used in a computation of the flow past cylinder along with two other seminal ideas: creation of vorticity on the boundary to enforce the no-slip boundary condition and using random walk to approximate the viscosity. Krasny performed careful studies for the vortex sheet problem with the vortex blob methods [15].…”

Section: Introductionmentioning

confidence: 99%

Convergence of the point vortex method for 2-D vortex sheet

Liu

Xin

2000

Math. Comp.

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Abstract. We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

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Section: Introductionmentioning

confidence: 99%

Convergence of the point vortex method for 2-D vortex sheet

Liu

Xin

2000

Math. Comp.

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“…The class of vortex methods are used to simulate incompressible and slightly compressible flows [41,65].…”

Section: Spectral Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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“…The origin of modern meshless methods could be dated back to the 70's with the pioneering works in generalized finite differences and vortex particle methods [20], [21], [22]. However, the strongest influence upon the present trends is commonly attributed to early Smoothed Particle Hydrodynamics (SPH) formulations [23], [24], [25], where a lagrangian particle tracking is used to describe the motion of a fluid.…”

Section: Meshless Approximantsmentioning

confidence: 99%

“…The overall accuracy of the resulting algorithm is, therefore, second order, and one integration point per cell face (the midpoint) is sufficient to compute the boundary integrals (Figure 2 left). Third order spatial accuracy could be obtained using the cubic basis (21). In this case a higher order quadrature rule is required for the evaluation of boundary integrals.…”

Section: Accuracy and Implementationmentioning

confidence: 99%

High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics

Cueto‐Felgueroso¹,

Colominas²,

Fe³

et al. 2005

Int. J. Numer. Meth. Engng

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SUMMARYThis paper introduces the use of Moving Least Squares (MLS) approximations for the development of high-order finite volume discretizations on unstructured grids. The field variables and their succesive derivatives can be accurately reconstructed using this meshfree technique in a general nodal arrangement. The methodology proposed is used in the construction of low-dissipative highorder high-resolution schemes for the shallow water equations. In particular, second and third-orderreconstruction upwind schemes for unstructured grids based on Roe's flux difference splitting are developed and applied to inviscid and viscous flows. This class of meshfree reconstruction techniques provide a robust and general approximation framework which represents an interesting alternative to the existing procedures, allowing, in addition, an accurate computation of the viscous fluxes. Copyright

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Numerical study of slightly viscous flow

Cited by 1,294 publications

References 7 publications

Convergence of the point vortex method for 2-D vortex sheet

Convergence of the point vortex method for 2-D vortex sheet

A review of numerical methods for nonlinear partial differential equations

High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics

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