Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations

Rider, William J.; Greenough, Jeffrey A.; Kamm, James R.

doi:10.1016/j.jcp.2007.02.023

Cited by 68 publications

(76 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In our computations, we take the one-dimensional delta function δ h (x) to be the four-point delta function of Peskin [1]. The motion of the Lagrangian mesh nodes is determined in the cell-centered case via 10) and in the staggered-grid case via …”

Section: Eulerian and Lagrangian Spatial Discretizationsmentioning

confidence: 99%

“…To compute the m th component of N = (N 1 , N 2 ), we first determine a staggered-grid advection velocity field u ADV,m that is defined on the edges of the control volumes associated with component m of the velocity field u. Next, we use the xsPPM7 variant [10] of the piecewise-parabolic method (PPM) [11] to determine u PPM m , an upwind-biased interpolation of u m , on the edges of the control volumes, using u ADV,m to determine the upwind direction. Finally, for each component m of the velocity field, we evaluate N m = u ADV,m ·∇ h u PPM m using second-order compact finite differences.…”

Section: Approximating the Convective Acceleration Term U·∇umentioning

confidence: 99%

“…We consider a collocated Eulerian discretization that approximates the pressure and the components of the velocity at the centers of the Cartesian grid cells, and a staggered-grid (i.e., marker-and-cell or MAC [9]) discretization that approximates the pressure at the centers of the Cartesian grid cells, and that approximates the velocity at the centers of the edges of the grid cells. We employ the same second-order accurate, semi-implicit time stepping scheme for both spatial discretizations, and we use nearly identical higher-order upwind discretizations for the convective acceleration term that are based on a version [10] of the piecewise-parabolic method [11].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Volume Conservation of the Immersed Boundary Method

Griffith

2012

Commun. comput. phys.

View full text Add to dashboard Cite

Abstract.The immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests considered herein, the staggered-grid IB scheme generally yields at least a modest improvement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of the collocated schemes. We also compare the performance of cell-centered schemes that use either exact or approximate projection methods. We find that the volumeconservation properties of approximate projection IB methods depend strongly on the formulation of the projection method. When used with the IB method, we find that pressure-free approximate projection methods can yield extremely poor volume conservation, whereas pressure-increment approximate projection methods yield volume conservation that is nearly identical to that of a cell-centered exact projection method.

show abstract

Section: Eulerian and Lagrangian Spatial Discretizationsmentioning

confidence: 99%

Section: Approximating the Convective Acceleration Term U·∇umentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Volume Conservation of the Immersed Boundary Method

Griffith

2012

Commun. comput. phys.

View full text Add to dashboard Cite

show abstract

“…Typically, these have been based on the idea allowing the representation of solution values outside the range defined by the cell averages [16], while still suppressing oscillations at discontinuities and underresolved gradients. In particular, the methods proposed to solve the problem to obtain uniform high-order accuracy for smooth solutions [6,8,7,13,2,10] typically have used quite elaborate analytic and / or geometric constructions. In this note, we propose a particularly simple approach to solving this problem for the PPM method [4].…”

Section: Introductionmentioning

confidence: 99%

A limiter for PPM that preserves accuracy at smooth extrema

Colella

Sekora

2008

Journal of Computational Physics

153

169

View full text Add to dashboard Cite

We present a new limiter for the PPM method of Colella and Woodward [4] that preserves accuracy at smooth extrema. It is based on constraining the interpolated values at extrema (and only at extrema) using nonlinear combinations of various difference approximations of the second derivatives. Otherwise, we use a standard geometric limiter to preserve monotonicity away from extrema. This leads to a method that has the same accuracy for smooth initial data as the underlying PPM method without limiting, while providing sharp, non-oscillatory representations of discontinuities.

show abstract

“…We now turn to gas dynamics codes which have a broad range of applications, from astrophysical and geophysical flow simulation to the design and performance analysis of engineering systems. Specifically, we discuss the validation of the "Cuervo" code developed at Los Alamos National Laboratory [105,106] for use as a simulation tool in the complex physics of compressible mixing. This software generates solutions of the Euler equations for flows of inviscid, non-heat-conducting, compressible gas.…”

Section: A Computational Fluid Dynamics Model For Shock-induced Mixingmentioning

confidence: 99%

A General Strategy for Physics-Based Model Validation Illustrated with Earthquake Phenomenology, Atmospheric Radiative Transfer, and Computational Fluid Dynamics

Sornette

Davis

Kamm

et al. 2008

Lecture Notes in Computational Science and Engineering

View full text Add to dashboard Cite

Summary. Validation is often defined as the process of determining the degree to which a model is an accurate representation of the real world from the perspective of its intended uses. Validation is crucial as industries and governments depend increasingly on predictions by computer models to justify their decisions. In this article, we survey the model validation literature and propose to formulate validation as an iterative construction process that mimics the process occurring implicitly in the minds of scientists. We thus offer a formal representation of the progressive build-up of trust in the model, and thereby replace incapacitating claims on the impossibility of validating a given model by an adaptive process of constructive approximation. This approach is better adapted to the fuzzy, coarse-grained nature of validation. Our procedure factors in the degree of redundancy versus novelty of the experiments used for validation as well as the degree to which the model predicts the observations. We illustrate the new methodology first with the maturation of Quantum Mechanics as the arguably best established physics theory and then with several concrete examples drawn from some of our primary scientific interests: a cellular automaton model for earthquakes, an anomalous diffusion model for solar radiation transport in the cloudy atmosphere, and a computational fluid dynamics code for the Richtmyer-Meshkov instability.

show abstract

Accurate monotonicity- and extrema-preserving methods through adaptive nonlinear hybridizations

Cited by 68 publications

References 42 publications

On the Volume Conservation of the Immersed Boundary Method

On the Volume Conservation of the Immersed Boundary Method

A limiter for PPM that preserves accuracy at smooth extrema

A General Strategy for Physics-Based Model Validation Illustrated with Earthquake Phenomenology, Atmospheric Radiative Transfer, and Computational Fluid Dynamics

Contact Info

Product

Resources

About