scholarly journals Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1930
Author(s):  
Zhen Yang ◽  
Junjie Ma
Keyword(s):  
Adaptive Mesh Refinement ◽  
Numerical Experiments ◽  
Singular Integrals ◽  
Fourier Transforms ◽  
Adaptive Mesh ◽  
Oscillatory Integrals ◽  
Efficient Computation ◽  
New Approach ◽  
Highly Oscillatory ◽  
Nearly Singular Integrals

In this paper, we consider fast and high-order algorithms for calculation of highly oscillatory and nearly singular integrals. Based on operators with regard to Chebyshev polynomials, we propose a class of spectral efficient Levin quadrature for oscillatory integrals over rectangle domains, and give detailed convergence analysis. Furthermore, with the help of adaptive mesh refinement, we are able to develop an efficient algorithm to compute highly oscillatory and nearly singular integrals. In contrast to existing methods, approximations derived from the new approach do not suffer from high oscillatory and singularity. Finally, several numerical experiments are included to illustrate the performance of given quadrature rules.

scholarly journals A new approach to modelling the two way coupling for momentum transfer in a hollow-cone spray

2017 ◽  
Author(s):  
Andreas Papoutsakis ◽  
Sergei Sazhin ◽  
Steven Begg ◽  
Ionut Danaila ◽  
Francky Luddens
Keyword(s):  
Carrier Phase ◽  
Adaptive Mesh Refinement ◽  
Energy Transport ◽  
Adaptive Mesh ◽  
Lagrangian Approach ◽  
New Approach ◽  
Computational Mesh ◽  
Initial Domain ◽  
Tree Data ◽  
Tree Data Structure

A new approach to modelling the interaction between droplets and the carrier phase is suggested. The new model isapplied to the analysis of a spray injected into a chamber of quiescent air, using an Eulerian-Lagrangian approach. The conservative formulation of the equations for mass, momentum and energy transport is used for the analysis of the carrier phase. The dispersed phase is modelled using the Lagrangian approach with droplets represented by individual parcels.The implementation of the Discontinuous Galerkin method (ForestDG), based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees, is used in our analysis. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution for the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of the accuracy of discretisation of shear layers and vortices.The initial domain consists of a graph of unitarian-trees representing hexahedral, prismatic and tetrahedral elements. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. These are attached to the tree data structure which facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree. This enables us to refine the computational mesh in the vicinity of the droplet parcels aiming to accurately resolve the coupling betweenthe two phases.DOI: http://dx.doi.org/10.4995/ILASS2017.2017.4671

A Multi-Fluid Investigation of the Membrane Supporting Grid Effects on the Richtmyer-Meshkov Instability

2019 ◽  
Vol 390 ◽  
pp. 1-7
Author(s):  
Mohamad Al-Marouf ◽  
Ravi Samtaney
Keyword(s):  
Adaptive Mesh Refinement ◽  
Numerical Experiments ◽  
Complex Geometry ◽  
Adaptive Mesh ◽  
Circular Cylinders ◽  
Small Scale ◽  
Gas Mixture ◽  
Material Interface ◽  
Wire Grid ◽  
Thin Wires

We present results of numerical experiments performed to evaluate the effects of the material interface supporting wire grid on the Richtmyer-Meshkov instability (RMI). An air-SF6 interface initially perturbed sinusoidally supported on a number of solid circular cylinders. These cylinders are introduced along the interface to mimic the presence of the grid thin wires. The resulted mixing and growth rate of the perturbation in the presence and absence of the supporting grid were analyzed and validated with experimental measurements. The small scales perturbation imposed by the cylinders are around two orders of magnitude smaller than the interface sinusoidal perturbation wavelength requiring the adaptive mesh refinement (AMR) to adequately resolve small scale features. Furthermore, an embedded boundary technique is used to handle the complex geometry stemming from the presence of these multiple. A multi-fluid formulation is utilized to form a multi-gas species interface and compute the gas mixture properties.

scholarly journals Efficient Computation of Highly Oscillatory Integrals by Using QTT Tensor Approximation

2016 ◽  
Vol 16 (1) ◽  
pp. 145-159 ◽  
Author(s):  
Boris Khoromskij ◽  
Alexander Veit
Keyword(s):  
Oscillatory Integrals ◽  
Integration Scheme ◽  
Efficient Computation ◽  
Oscillatory Function ◽  
Numerical Integration Scheme ◽  
Wide Range ◽  
Spatial Dimensions ◽  
Highly Oscillatory ◽  
Highly Oscillatory Integrals ◽  
Logarithmic Complexity

AbstractWe propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter ${\omega \ge 0}$, typically varying in a large interval. Our approach is based, for a fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain ω-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals which makes them difficult to evaluate. Furthermore, they have to be approximated typically in large intervals. Here we use the quantized-tensor train (QTT) approximation method for functional M-vectors of logarithmic complexity in M in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions.

Adaptive Mesh Refinement in 2D – An Efficient Implementation in Matlab

2020 ◽  
Vol 20 (3) ◽  
pp. 459-479 ◽  
Author(s):  
Stefan A. Funken ◽  
Anja Schmidt
Keyword(s):  
Data Structure ◽  
Adaptive Mesh Refinement ◽  
Numerical Experiments ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Two Dimensions ◽  
Efficient Implementation ◽  
Research And Education ◽  
Mesh Refinements ◽  
Implementation In Matlab

AbstractThis paper deals with the efficient implementation of various adaptive mesh refinements in two dimensions in Matlab. We give insights into different adaptive mesh refinement strategies allowing triangular and quadrilateral grids with and without hanging nodes. Throughout, the focus is on an efficient implementation by utilization of reasonable data structure, use of Matlab built-in functions and vectorization. This paper shows the transition from theory to implementation in a clear way and thus is meant to serve educational purposes of how to implement a method while keeping the code as short as possible – an implementation of an efficient adaptive mesh refinement is possible within 71 lines of Matlab. Numerical experiments underline the efficiency of the code and show the flexible deployment in different contexts where adaptive mesh refinement is in use. Our implementation is accessible and easy-to-understand and thus considered to be a valuable tool in research and education.

scholarly journals Stable Dynamical Adaptive Mesh Refinement

2021 ◽  
Vol 86 (3) ◽  
Author(s):  
Tomas Lundquist ◽  
Jan Nordström ◽  
Arnaud Malan
Keyword(s):  
Finite Difference ◽  
Adaptive Mesh Refinement ◽  
Numerical Experiments ◽  
Adaptive Mesh ◽  
Time Dependent ◽  
Inner Product ◽  
Finite Difference Schemes ◽  
Adaptive Meshes ◽  
Moving Interfaces ◽  
Interpolation Operators

AbstractWe consider accurate and stable interpolation procedures for numerical simulations utilizing time dependent adaptive meshes. The interpolation of numerical solution values between meshes is considered as a transmission problem with respect to the underlying semi-discretized equations, and a theoretical framework using inner product preserving operators is developed, which allows for both explicit and implicit implementations. The theory is supplemented with numerical experiments demonstrating practical benefits of the new stable framework. For this purpose, new interpolation operators have been designed to be used with multi-block finite difference schemes involving non-collocated, moving interfaces.

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