55 research outputs found
Complete Algebraic Reconstruction of Piecewise-Smooth Functions from Fourier Data
In this paper we provide a reconstruction algorithm for piecewise-smooth
functions with a-priori known smoothness and number of discontinuities, from
their Fourier coefficients, posessing the maximal possible asymptotic rate of
convergence -- including the positions of the discontinuities and the pointwise
values of the function. This algorithm is a modification of our earlier method,
which is in turn based on the algebraic method of K.Eckhoff proposed in the
1990s. The key ingredient of the new algorithm is to use a different set of
Eckhoff's equations for reconstructing the location of each discontinuity.
Instead of consecutive Fourier samples, we propose to use a "decimated" set
which is evenly spread throughout the spectrum
HIGH ORDER SHOCK CAPTURING SCHEMES FOR HYPERBOLIC CONSERVATION LAWS AND THE APPLICATION IN OPEN CHANNEL FLOWS
Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows
Steady-state response of a random dynamical system described with Padé approximants and random eigenmodes
Designing a random dynamical system requires the prediction of the statistics of the response, knowing the random model of the uncertain parameters. Direct Monte Carlo simulation (MCS) is the reference method for propagating uncertainties but its main drawback is the high numerical cost. A surrogate model based on a polynomial chaos expansion (PCE) can be built as an alternative to MCS. However, some previous studies have shown poor convergence properties around the deterministic eigenfrequencies. In this study, an extended Pade approximant approach is proposed not only to accelerate the convergence of the PCE but also to have a better representation of the exact frequency response, which is a rational function of the uncertain parameters. A second approach is based on the random mode expansion of the response, which is widely used for deterministic dynamical systems. A PCE approach is used to calculate the random modes. Both approaches are tested on an example to check their efficiency
A new iterative approach to solving the transport equation
We present a new iterative approach to solving neutral-particle transport
problems. The scheme divides the transport solution into its particular and
homogeneous or “source-free” components. The particular problem is solved directly,
while the homogeneous problem is found iteratively. To organize the iterative inversion
of the homogeneous components, we exploit the structures of the so called Case-modes
that compose it. The asymptotic Case-modes, those that vary slowly in space and angle,
are assigned to a diffusion solver. The remaining transient Case-modes, those with large
spatial gradients, are assigned to a transport solver. The scheme iterates on the
contribution from each solver until the particular plus homogeneous solution converges.
The iterative method is implemented successfully in slab geometry with isotropic
scattering and one energy group. The convergence rate of the method is only weakly
dependent on the scattering ratio of the problem. Instead, the rate of convergence
depends strongly on the material thickness of the slab, with thick slabs converging in
few iterations. The transient solution is obtained by applying a One Cell Inversion
scheme instead of a Source Iteration based scheme. Thus, the transient unknowns are
calculated with little coordination between them. This independence among unknowns
makes our scheme ideally suited for transport calculations on parallel architectures.
The slab geometry iterative scheme is adapted to XY geometry. Unfortunately,
this attempt to extend the slab geometry iterative scheme to multiple dimensions has not
been successful. The exact filtering scheme needed to discriminate asymptotic and
transient modes has not been obtained and attempts to approximate this filtering process resulted in a divergent iterative scheme. However, the development of this iterative
scheme yield valuable analysis tools to understand the Case-mode structure of any
spatial discretization under arbitrary material properties
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Numerical Analysis of Flux Reconstruction
High-order methods have become of increasing interest in recent years in computational
physics. This is in part due to their perceived ability to, in some cases, reduce the computational overhead of complex problems through both an efficient use of computational
resources and a reduction in the required degrees of freedom. One such high-order
method in particular – Flux Reconstruction – is the focus of this thesis. This body of work
relies and expands on the theoretical methods that are used to understand the behaviour
of numerical methods – particularly related to their real-world application to industrial
problems.
The thesis begins by challenging some of the existing dogma surrounding computational fluid dynamics by evaluating the performance of high-order flux reconstruction.
First, the use of the primitive variables as an intermediary step in the construction of flux
terms is investigated. It is found that reducing the order of the flux function by using the
conserved rather than primitive variables has a substantial impact on the resolution of
the method. Critically, this is supported by a theoretical analysis, which shows that this
mechanism of error generation becomes increasing important to consider as the order of
accuracy increases.
Next, the analysis of Flux Reconstruction was extended by analytically and numerically exploring the impact of higher dimensionality and grid deformation. It is found
that both expanding and contracting grids – essential components of real-world domain
decomposition – can cause dispersion overshoot in two dimensions, but that FR appears
to suffer less that comparable Finite Difference approaches. Fully discrete analysis is then
used to show that, depending on the correction function, small perturbations in incidence
angle can cause large changes in group velocity. The same analysis is also used to theoretically demonstrate that Discontinuous Galerkin suffers less from dispersion error than
Huynh’s FR scheme – a phenomenon that has previously been observed experimentally,
but not explained theoretically.
This thesis concludes with the presentation of a robust theoretical underpinning for
determining stable correction functions for FR. Three new families of correction functions
are presented, and their properties extensively explored. An important theoretical finding
is introduced – that stable correction functions are not defined uniquely be a norm. As a
result, a generalised approach is presented, which is able to recover all previously defined
correction functions, but in some instances via a different norm to their original derivation.
This new super-family of correction functions shows considerable promise in increasing
temporal stability limits, reducing dispersion when fully discretised, and increasing the
rate of convergence.
Taken altogether, this thesis represents a considerable advance in the theoretical
characterisation and understanding of a numerical method – one which, it has been shown,
has enormous potential for forming the heart of future computational physics codes
Recent Advances in Industrial and Applied Mathematics
This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress
Recent Advances in Industrial and Applied Mathematics
This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress
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