<i>H</i>-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

Berger, Marsha; Helzel, Christiane; LeVeque, Randall J.

doi:10.1137/s0036142902405394

Cited by 85 publications

(75 citation statements)

References 29 publications

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“…This is the well-known small-cell problem for embedded boundary methods. There have been a number of proposals to deal with this problem, including merging the small control volumes with nearby larger ones [13,6], and the development of specialized stencils that guarantee the required cancellations in (8) [4,3]. The approach we have taken to this problem has been to expand the range of influence of the small control volumes algebraically to obtain a stable method [5,2,11].…”

Section: Stable Evolution Of Hyperbolic Conservation Lawsmentioning

confidence: 99%

A Cartesian grid embedded boundary method for hyperbolic conservation laws

Colella

Graves

Keen

et al. 2006

Journal of Computational Physics

186

177

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We present a second-order Godunov algorithm to solve time-dependent hyperbolic systems of conservation laws on irregular domains. Our approach is based on a formally consistent discretization of the conservation laws on a finite-volume grid obtained from intersecting the domain with a Cartesian grid. We address the smallcell stability problem associated with such methods by hybridizing our conservative discretization with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation. The resulting method is second-order accurate in L 1 for smooth problems, and is robust in the presence of large-amplitude discontinuities intersecting the irregular boundary.

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Section: Stable Evolution Of Hyperbolic Conservation Lawsmentioning

confidence: 99%

A Cartesian grid embedded boundary method for hyperbolic conservation laws

Colella

Graves

Keen

et al. 2006

Journal of Computational Physics

186

177

View full text Add to dashboard Cite

show abstract

“…For example, Pember et al [5] used a Cartesian grid method for solving the time-dependent equations of gas dynamics. For the one and two-dimensional Euler equations, Berger, Helzel and LeVeque [6] developed a Cartesian "h-box" method which aims at avoiding the small-cell time step restriction without sacrificing accuracy. Zhang and LeVeque [7] solved the acoustic wave equation with discontinuous coefficients written as a first order system.…”

Section: Introductionmentioning

confidence: 99%

A Second Order Accurate Embedded Boundary Method for the Wave Equation with Dirichlet Data

Kreiss¹,

Petersson²

2006

SIAM J. Sci. Comput.

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The accuracy of Cartesian embedded boundary methods for the second order wave equation in general two-dimensional domains subject to Dirichlet boundary conditions is analyzed. Based on the analysis, we develop a numerical method where both the solution and its gradient are second order accurate. We avoid the small-cell stiffness problem without sacrificing the second order accuracy by adding a small artificial term to the Dirichlet boundary condition. Long-time stability of the method is obtained by adding a small fourth order dissipative term. Several numerical examples are provided to demonstrate the accuracy and stability of the method. The method is also used to solve the two-dimensional TM z problem for Maxwell's equations posed as a second order wave equation for the electric field coupled to ordinary differential equations for the magnetic field.

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“…We have worked on the development of such methods in the past (e.g., [5], [6], [9], [11], [13], [25], [35]) and believe that it is a good approach for complex geometries. This approach has been advanced by many other researchers as well, for hyperbolic equations (e.g., [16], [17], [19], [20]), elliptic equations (e.g., [27], [38], [39]), and incompressible Navier-Stokes equations (e.g., [2], [32]).…”

mentioning

confidence: 99%

Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains

Calhoun¹,

Helzel²,

LeVeque³

2008

SIAM Rev.

Self Cite

122

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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 three-dimensional 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 non-orthogonal, we show that the high-resolution wave-propagation 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 latitude-longitude 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 reaction-diffusion 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.Key words. circular domain, spherical domain, grid transformations, finite volume methods, hyperbolic PDEs AMS subject classifications. 65M06, 65M50, 65Y15, 35L60, 58J451. Introduction. Uniform Cartesian grids are well suited to solving partial differential equations in rectangular regions. Logically rectangular quadrilateral grids work well for many problems where the domain is a deformed rectangle. For example, in an annulus a rectangular grid in polar r-θ coordinates can be used with periodic boundary conditions in the θ direction.It is not so clear that quadrilateral grids are appropriate for solving problems in smooth domains such as a circle (i.e., a planar disk). Polar coordinates could again be used over a rectangular computational domain, but when polar coordinates are used over the full circle there may be numerical difficulties arising at the center of the circle where all of the radial grid lines meet at a single point in physical space. While it is possible to use finite volume methods in this context, these converging grid lines give rise to cells near the center that are much smaller than cells elsewhere in the domain. For explicit finite volume methods the CFL condition may then require a very small time step everywhere. Similar problems arise when using a latitude-longitude grid on the sphere.In this paper we consider a family of quadrilateral and hexahedral grids for the numerical solut...

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H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

Cited by 85 publications

References 29 publications

A Cartesian grid embedded boundary method for hyperbolic conservation laws

A Cartesian grid embedded boundary method for hyperbolic conservation laws

A Second Order Accurate Embedded Boundary Method for the Wave Equation with Dirichlet Data

Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains

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