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14
Immersed Interface Methods For Stokes Flow With Elastic Boundaries Or Surface Tension
 SIAM J. Sci. Comput
"... . A second order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interf ..."
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Cited by 75 (13 self)
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. A second order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in a previous paper (SIAM J. Numer. Anal., 31(1994), pp. 10191044). The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasiNewton method is developed that allows reasonable time steps to be used. Key words. Stokes flow, creeping flow, interface tracking, discontinuous coefficients, immersed interface methods, Cartesian grids, bubbles. AMS subject clas...
Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
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An Adaptive Cartesian Grid Method For Unsteady Compressible Flow In Irregular Regions
 J. Comput. Phys
, 1993
"... In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit secondorder Godunov algorithm fo ..."
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Cited by 48 (14 self)
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In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit secondorder Godunov algorithm followed by a corrector applied to cells at the boundary. The discretization near the fluidbody interface is based on a volumeoffluid approach with a redistribution procedure to maintain conservation while avoiding time step restrictions arising from small cells where the boundary intersects the mesh. The single grid Cartesian mesh integration scheme is coupled to a conservative adaptive mesh refinement algorithm that selectively refines regions of the computational grid to achieve a desired level of accuracy. Examples showing the results of the combined Cartesian grid integration/adaptive mesh refinement algorithm for both two and threedimensional flows are presented. (This page intent...
A Cartesian grid projection method for the incompressible Euler equations in complex geometries
 SIAM J. Sci. Comput
, 1998
"... Abstract. Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating timedependent incompressible inviscid flow which combines a projection method with a “Cartesian grid ” approach for representing ..."
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Cited by 29 (6 self)
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Abstract. Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating timedependent incompressible inviscid flow which combines a projection method with a “Cartesian grid ” approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volumeoffluid representation. A redistribution procedure is used to eliminate timestep restrictions due to small cells where the boundary intersects the mesh. The projection step uses an approximate projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and for the timedependent algorithm in two dimensions. The method is also demonstrated on flow past a halfcylinder with vortex shedding. Key words. Cartesian grid, projection method, incompressible Euler equations
A triangular cutcell adaptive method for highorder discretizations of the compressible Navier–Stokes equations
, 2007
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A Genuinely Multidimensional Upwind Scheme and an Efficient Multigrid for the Compressible Euler Equations
, 1994
"... We present a new approach towards the construction of a genuinely multidimensional highresolution scheme for computing steadystate solutions of the Euler equations of gas dynamics. The unique advantage of this approach is that the GaussSeidel relaxation is stable when applied directly to the high ..."
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Cited by 15 (3 self)
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We present a new approach towards the construction of a genuinely multidimensional highresolution scheme for computing steadystate solutions of the Euler equations of gas dynamics. The unique advantage of this approach is that the GaussSeidel relaxation is stable when applied directly to the highresolution discrete equations, thus allowing to obtain a very good efficiency of the multigrid steadystate solver. This is the only highresolution scheme known to us that has this property. The twodimensional scheme is presented in details. It is formulated on triangular (structured and unstructured) meshes and can be interpreted as a genuinely twodimensional extension of the Roe scheme. The quality of the solutions obtained using this scheme and the performance of the multigrid algorithm are illustrated by the numerical experiments. Construction of the threedimensional scheme is outlined briefly as well.
A Cartesian Grid FiniteVolume Method for the AdvectionDiffusion Equation in Irregular Geometries
 J. COMPUT. PHYS
, 1999
"... We present a fully conservative, highresolution, finite volume algorithm for advectiondiffusion equations in irregular geometries. The algorithm uses a Cartesian grid in which some cells are cut by the embedded boundary. A novel feature is the use of a "capacity function" to model the fact that so ..."
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Cited by 9 (2 self)
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We present a fully conservative, highresolution, finite volume algorithm for advectiondiffusion equations in irregular geometries. The algorithm uses a Cartesian grid in which some cells are cut by the embedded boundary. A novel feature is the use of a "capacity function" to model the fact that some cells are only partially available to the fluid. The advection portion then uses the explicit wavepropagation methods implemented in clawpack, and is stable for Courant numbers up to 1. Diffusion is modelled with an implicit finite volume algorithm. Results are shown for several geometries. Convergence if verified and the 1norm order of accuracy is found to between 1.2 and 2 depending on the geometry and Peclet number. Software is available on the web.
Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
 In preparation; http://www.amath. washington.edu/~rjl/pubs/circles
, 2005
"... Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational do ..."
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Cited by 8 (3 self)
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Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the highresolution wavepropagation algorithm implemented in clawpack can be effectively used to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grid is below 2 for most of our grid mappings, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitudelongitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reactiondiffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with matlab routines for the various mappings.
Adaptive Parallel Meshes with Complex Geometry
, 1991
"... We discuss the automatic creation and adaptive refinement of an unstructured mesh within a complex geometry such as the space surrounding an airplane. This may be formulated as two distinct parts; a nonparallel part requiring global knowledge which automatically creates a coarse compatible mesh, an ..."
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Cited by 7 (0 self)
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We discuss the automatic creation and adaptive refinement of an unstructured mesh within a complex geometry such as the space surrounding an airplane. This may be formulated as two distinct parts; a nonparallel part requiring global knowledge which automatically creates a coarse compatible mesh, and a parallel local refinement algorithm, which refines the mesh until simulation can begin, then adaptively refines it according to the progress of the simulation.