189 research outputs found
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A posteriori corrections for cubic and quintic interpolating splines at equally-spaced knots
The method proposed recently by Lucas [13], for the a posteriori correction of odd degree interpolating periodic splines is extended to non-periodic cubic and quintic splines
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Superconvergence properties of quintic interpolatroy splines
Let Q be a quintic spline with equi-spaced knots on [a,b] interpolating a given function y at the knots. The parameters which determine Q are used to construct a piecewise defined polynomial P of degree six. It is shown that P can be used to give at any point of [a,b] better orders of approximation to y and its derivatives than those obtained from Q. It is also shown that the superconvergence properties of the derivatives of Q, at specific points of [a,b], are all simple consequences of the properties of P
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A class of cubic and quintic spline modified collocation methods for the solution of two-point boundary value problems
This paper is concerned with the study of a class of methods for solving second and fourth-order two-point boundary-value problems. The methods under
consideration are modifications of the standard cubic and quintic spline
collocation techniques, and are derived by making use of recent results con- cerning the a posteriori correction of cubic and quintic interpolating spline
Lacunary interpolation by quartic splines on uniform meshes
AbstractThe interpolation of a discrete set of data on the interval [0, 1], representing the first and the second derivatives (except at 0) of a smooth function f is investigated via quartic C2-splines. Error bounds in the uniform norm for ∥s(i) − f(i)∥, i=0(1)2, if f ∈ Cl[0, 1], l=3, 5 and (3) ∈ BV[0, 1], together with computational examples will also be presented
Geometry considerations for high-order finite-volume methods on structured grids with adaptive mesh refinement
2022 Summer.Includes bibliographical references.Computational fluid dynamics (CFD) is an invaluable tool for engineering design. Meshing complex geometries with accuracy and efficiency is vital to a CFD simulation. In particular, using structured grids with adaptive mesh refinement (AMR) will be invaluable to engineering optimization where automation is critical. For high-order (fourth-order and above) finite volume methods (FVMs), discrete representation of complex geometries adds extra challenges. High-order methods are not trivially extended to complex geometries of engineering interest. To accommodate geometric complexity with structured AMR in the context of high-order FVMs, this work aims to develop three new methods. First, a robust method is developed for bounding high-order interpolations between grid levels when using AMR. High-order interpolation is prone to numerical oscillations which can result in unphysical solutions. To overcome this, localized interpolation bounds are enforced while maintaining solution conservation. This method provides great flexibility in how refinement may be used in engineering applications. Second, a mapped multi-block technique is developed, capable of representing moderately complex geometries with structured grids. This method works with high-order FVMs while still enabling AMR and retaining strict solution conservation. This method interfaces with well-established engineering work flows for grid generation and interpolates generalized curvilinear coordinate transformations for each block. Solutions between blocks are then communicated by a generalized interpolation strategy while maintaining a single-valued flux. Finally, an embedded-boundary technique is developed for high-order FVMs. This method is particularly attractive since it automates mesh generation of any complex geometry. However, the algorithms on the resulting meshes require extra attention to achieve both stable and accurate results near boundaries. This is achieved by performing solution reconstructions using a weighted form of high-order interpolation that accounts for boundary geometry. These methods are verified, validated, and tested by complex configurations such as reacting flows in a bluff-body combustor and Stokes flows with complicated geometries. Results demonstrate the new algorithms are effective for solving complex geometries at high-order accuracy with AMR. This study contributes to advance the geometric capability in CFD for efficient and effective engineering applications
Theory-based Benchmarking of the Blended Force-Based Quasicontinuum Method
We formulate an atomistic-to-continuum coupling method based on blending
atomistic and continuum forces. Our precise choice of blending mechanism is
informed by theoretical predictions. We present a range of numerical
experiments studying the accuracy of the scheme, focusing in particular on its
stability. These experiments confirm and extend the theoretical predictions,
and demonstrate a superior accuracy of B-QCF over energy-based blending
schemes.Comment: 25 pages, color figures; some numerical experiments re-don
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