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15
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 22 (6 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.
A highresolution rotated grid method for conservation laws with embedded geometries
 SIAM J. Sci. Comput
"... Abstract. We develop a secondorder rotated grid method for the approximation of time dependent solutions of conservation laws in complex geometry using an underlying Cartesian grid. Stability for time steps adequate for the regular part of the grid is obtained by increasing the domain of dependence ..."
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Cited by 12 (3 self)
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Abstract. We develop a secondorder rotated grid method for the approximation of time dependent solutions of conservation laws in complex geometry using an underlying Cartesian grid. Stability for time steps adequate for the regular part of the grid is obtained by increasing the domain of dependence of the numerical method near the embedded boundary by constructing hboxes at grid cell interfaces. We describe a construction of hboxes that not only guarantees stability but also leads to an accurate and conservative approximation of boundary cells that may be orders of magnitude smaller than regular grid cells. Of independent interest is the rotated difference scheme itself, on which the embedded boundary method is based.
Error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation
 SIAM J. Numer. Anal
, 2005
"... Abstract. This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in R nd and every non vani ..."
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Cited by 8 (1 self)
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Abstract. This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in R nd and every non vanishing vector a of R nd. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs. Then we prove that this norm is indeed bounded by the mesh size in various cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimension. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases like the one with independent refined meshes.
3D adaptive central schemes: part I. Algorithms for assembling the dual mesh
 Appl Numer Math
"... Abstract. Central schemes are frequently used for incompressible and compressible flow calculations. The present paper is the first in a forthcoming series where a new approach to a 2nd order accurate Finite Volume scheme operating on cartesian grids is discussed. Here we start with an adaptively re ..."
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Cited by 7 (0 self)
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Abstract. Central schemes are frequently used for incompressible and compressible flow calculations. The present paper is the first in a forthcoming series where a new approach to a 2nd order accurate Finite Volume scheme operating on cartesian grids is discussed. Here we start with an adaptively refined cartesian primal grid in 3D and present a construction technique for the staggered dual grid based on L ∞Voronoi cells. The local refinement constellation on the primal grid leads to a finite number of uniquely defined local patterns on a primal cell. Assembling adjacent local patterns forms the dual grid. All local patterns can be analysed in advance. Later, running the numerical scheme on staggered grids, all necessary geometric information can instantly be retrieved from lookuptables. The new scheme is compared to established ones in terms of algorithmic complexity and computational effort.
A simple second order cartesian scheme for compressible Euler flows
, 2012
"... We present a finitevolume scheme for compressible Euler flows where the grid is cartesian and it does not fit to the body. The scheme, based on the definition of an ad hoc Riemann problem at solid boundaries, is simple to implement and it is formally second order accurate. Error convergence rates w ..."
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Cited by 5 (1 self)
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We present a finitevolume scheme for compressible Euler flows where the grid is cartesian and it does not fit to the body. The scheme, based on the definition of an ad hoc Riemann problem at solid boundaries, is simple to implement and it is formally second order accurate. Error convergence rates with respect to several exact test cases are investigated and examples of flow solutions in one, two and three dimensions are presented. 1
An embedded boundary method for the NavierStokes equations on a timedependent domain
 Comm. App. Math. Comp. Sci
"... mathematical sciences publishersCOMM. APP. MATH. AND COMP. SCI. ..."
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Cited by 4 (0 self)
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mathematical sciences publishersCOMM. APP. MATH. AND COMP. SCI.
An embedded boundary method for the wave equation with discontinuous coefficients
 SIAM J. Sci. Comput
"... Abstract A second order accurate embedded boundary method for the twodimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is al ..."
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Cited by 2 (0 self)
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Abstract A second order accurate embedded boundary method for the twodimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is allowed to intersect the mesh in an arbitrary fashion. By using ghost points on either side of the interface, previous embedded boundary techniques for the Neumann and Dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy. The resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points, resulting in a fully explicit timeintegration method. Numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder. The numerical solutions are evaluated against the analytical solution due to Mie, and pointwise second order accuracy is confirmed.
ASIMPLIFIEDhBOX METHOD FOR EMBEDDED BOUNDARY GRIDS ∗
"... Abstract. We present a simplified hbox method for integrating timedependent conservation laws on embedded boundary grids using an explicit finite volume scheme. By using a method of lines approach with a strong stability preserving Runge–Kutta method in time, the complexity of our previously intro ..."
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Abstract. We present a simplified hbox method for integrating timedependent conservation laws on embedded boundary grids using an explicit finite volume scheme. By using a method of lines approach with a strong stability preserving Runge–Kutta method in time, the complexity of our previously introduced hbox method is greatly reduced. A stable, accurate, and conservative approximation is obtained by constructing a finite volume method where the numerical fluxes satisfy acertaincancellationproperty. Foramodelprobleminonespacedimensionusingappropriate limiting strategies, the resulting method is shown to be total variation diminishing. In two space dimensions, stability is maintained by using rotated hboxes as introduced in previous work [M. J.
unknown title
, 2005
"... www.elsevier.com/locate/jcp A Cartesian grid embedded boundary method for hyperbolic conservation laws ..."
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www.elsevier.com/locate/jcp A Cartesian grid embedded boundary method for hyperbolic conservation laws