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Gerris: A TreeBased Adaptive Solver For The Incompressible Euler Equations In Complex Geometries
 J. Comp. Phys
, 2003
"... An adaptive mesh projection method for the timedependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volumeoffluid approach. Sec ..."
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Cited by 42 (10 self)
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An adaptive mesh projection method for the timedependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volumeoffluid approach. Secondorder convergence in space and time is demonstrated on regular, statically and dynamically refined grids. The quad/octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features. The source code of the method implementation is freely available.
A cellcentered adaptive projection method for the incompressible NavierStokes equations
"... We present an algorithm to compute adaptive solutions for incompressible flows using blockstructured local refinement in both space and time. This method uses a projection formulation based on a cellcentered approximate projection, which allows the use of a single set of cellcentered solvers. Bec ..."
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Cited by 33 (10 self)
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We present an algorithm to compute adaptive solutions for incompressible flows using blockstructured local refinement in both space and time. This method uses a projection formulation based on a cellcentered approximate projection, which allows the use of a single set of cellcentered solvers. Because of refinement in time, additional steps are taken to accurately discretize the advection and projection operators at grid refinement boundaries using composite operators which span the coarse and refined grids. This ensures that the method is approximately freestream preserving and satisfies an appropriate form of the divergence constraint. c ○ 2000 Academic Press Key Words: adaptive mesh refinement; incompressible flow; projection methods. 1.
On the Stability of GodunovProjection Methods for Incompressible Flow
, 1995
"... An analysis of the stability of certain numerical methods for the linear advectiondiffusion equation in two dimensions is performed. The advectiondiffusion equation is studied because it is a linearized version of the NavierStokes equations, the evolution equation for density in Boussinesq flows, ..."
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Cited by 19 (0 self)
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An analysis of the stability of certain numerical methods for the linear advectiondiffusion equation in two dimensions is performed. The advectiondiffusion equation is studied because it is a linearized version of the NavierStokes equations, the evolution equation for density in Boussinesq flows, and a simplified form of the equations for bulk thermodynamic temperature and mass fraction in reacting flows. It is found that various methods currently in use which are based on a CrankNicholson type temporal discretization utilizing secondorder Godunov methods for explicitly calculating advective terms suffer from a timestep restriction which depends on the coefficients of diffusive terms. A simple modification in the computation of the advective derivatives results in a method with a stability condition that is independent of the magnitude of the coefficients of the diffusive terms.
Approximate Projection Methods: Part I. Inviscid Analysis
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 1999
"... The use of approximate projection methods for modeling low Mach number ows avoids many of the numerical complications associated with exact projection methods, but introduces additional design choices in developing a robust algorithm. In this paper we rst explore these design choices in the setting ..."
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Cited by 15 (1 self)
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The use of approximate projection methods for modeling low Mach number ows avoids many of the numerical complications associated with exact projection methods, but introduces additional design choices in developing a robust algorithm. In this paper we rst explore these design choices in the setting of inviscid incompressible ow using several computational examples. We then develop a framework for analyzing the behavior of the dierent design variations and use that analysis to explain the features observed in the computations. As part of this work we introduce a new variation of the approximate projection algorithm that combines the advantages of several of the previous versions.
The lattice Boltzmann equation method: Theoretical interpretation, numerics and implications
 Int. Multiphase Flow, J
, 2003
"... During the last ten years the Lattice Boltzmann Equation (LBE) method has been developed as an alternative numerical approach in computational fluid dynamics (CFD). Originated from the discrete kinetic theory, the LBE method has emerged with the promise to become a superior modeling platform, both c ..."
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Cited by 14 (0 self)
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During the last ten years the Lattice Boltzmann Equation (LBE) method has been developed as an alternative numerical approach in computational fluid dynamics (CFD). Originated from the discrete kinetic theory, the LBE method has emerged with the promise to become a superior modeling platform, both computationally and conceptually, compared to the existing arsenal of the continuumbased CFD methods. The LBE method has been applied for simulation of various kinds of fluid flows under different conditions. The number of papers on the LBE method and its applications continues to grow rapidly, especially in the direction of complex and multiphase media. The purpose of the present paper is to provide a comprehensive, selfcontained and consistent tutorial on the LBE method, aiming to clarify misunderstandings and eliminate some confusion that seems to persist in the LBErelated CFD literature. The focus is placed on the fundamental principles of the LBE approach. An excursion into the history, physical background and details of the theory and numerical implementation is made. Special attention is paid to advantages and limitations of the method, and its perspectives to be a useful framework for description of complex flows and interfacial (and multiphase) phenomena. The computational performance of the LBE method is examined, comparing it to other CFD methods, which directly solve for the transport equations of the macroscopic variables.
Perspective on Eulerian Finite Volume Methods for Incompressible Interfacial Flows
 Kuhlmann and H Rath
, 1999
"... Incompressible interfacial flows here refer to those incompressible flows possessing multiple distinct, immiscible fluids separated by interfaces of arbitrarily complex topology. A prototypical example is free surface flows, where fluid properties across the interface vary by orders of magnitude. In ..."
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Cited by 6 (1 self)
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Incompressible interfacial flows here refer to those incompressible flows possessing multiple distinct, immiscible fluids separated by interfaces of arbitrarily complex topology. A prototypical example is free surface flows, where fluid properties across the interface vary by orders of magnitude. Interfaces present in these flows possess topologies that are not only irregular but also dynamic, undergoing gross changes such as merging, tearing, and filamenting as a result of the flow and interface physics such as surface tension and phase change. The interface topology requirements facing an algorithm tasked to model these flows inevitably leads to an underlying Eulerian methodology. The discussion herein is confined therefore to Eulerian schemes, with further emphasis on finite volume methods of discretization for the partial differential equations manifesting the physical model. Numerous algorithm choices confront users and developers of simulation tools designed to model the timeun...