In this work we consider a coupled system of m(>= 2) linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms with discontinuous source term. The leading term of each equation ...
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ISBN:
(纸本)9783319257273;9783319257259
In this work we consider a coupled system of m(>= 2) linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms with discontinuous source term. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct in magnitude. Overlapping boundary and interior layers can appear in the solution. A numerical method is constructed that involve an appropriate piecewise-uniform Shishkin mesh, which is fitted to both the boundary and interior layers. The parameter-uniform convergence of the numerical approximations is examined.
We examine the use of Hermite interpolation, that is interpolation using derivative data, in place of Lagrange interpolation to develop high-order PDE solvers. The fundamental properties of Hermite interpolation are r...
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ISBN:
(纸本)9783319198002;9783319197999
We examine the use of Hermite interpolation, that is interpolation using derivative data, in place of Lagrange interpolation to develop high-order PDE solvers. The fundamental properties of Hermite interpolation are recalled, with an emphasis on their smoothing effect and robust performance for nonsmooth functions. Examples from the CHIDES library are presented to illustrate the construction and performance of Hermite methods for basic wave propagation problems.
We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for stiff unsteady flows with boundary layers. As a model problem for the Navier-Stokes equations we consid...
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ISBN:
(纸本)9783319198002;9783319197999
We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for stiff unsteady flows with boundary layers. As a model problem for the Navier-Stokes equations we consider a two-dimensional advection-diffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summation-by-parts operators, and compare the results to an existing popular fourth order diagonally implicit Runge-Kutta method. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping.
We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analysis (IGA). The input into our pipeline is a boundary represented 3D model, either as a triangulation or as a collection ...
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ISBN:
(数字)9783319233154
ISBN:
(纸本)9783319233154;9783319233147
We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analysis (IGA). The input into our pipeline is a boundary represented 3D model, either as a triangulation or as a collection of trimmed non-uniform rational B-spline (NURBS) surfaces. The pipeline consists of three stages: computer aided design (CAD) model reconstruction from a triangulation (if necessary);segmentation of the boundary-represented solid into topological hexahedra;and volume parameterization. The result is a collection of volumetric NURBS patches. In this paper we discuss our methods for the three stages, and demonstrate the suitability of the result for IGA by performing stress simulations with examples of the output.
It has been noted in the past that discontinuous Galerkin methods can be viewed as a low order multi-domain Spectral method with penalty term (Hesthaven et al., Spectral methods for time-dependent problems, Cambridge ...
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ISBN:
(纸本)9783319198002;9783319197999
It has been noted in the past that discontinuous Galerkin methods can be viewed as a low order multi-domain Spectral method with penalty term (Hesthaven et al., Spectral methods for time-dependent problems, Cambridge University Press, Cambridge, 2007). It is then logical to first ask how to relate filters in Spectral Methods to Smoothness-Increasing Accuracy-Conservin (SIAC) filters, which are typically applied to approximations obtained via the discontinuous Galerkin methods. In this article we make a first effort to relate Smoothness-Increasing Accuracy-Conserving filtering to filtering for Spectral Methods. We frame this discussion in the context of Vandeven (J Sci Comput 6: 159-192, 1991).
Isogeometric Analysis (IgA) uses the same class of basis functions for both representing the geometry of the computational domain and approximating the solution of the boundary value problem under consideration. In pr...
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ISBN:
(数字)9783319233154
ISBN:
(纸本)9783319233154;9783319233147
Isogeometric Analysis (IgA) uses the same class of basis functions for both representing the geometry of the computational domain and approximating the solution of the boundary value problem under consideration. In practical applications, geometrical patches are used in order to get flexibility in the geometrical representation. This multi-patch representation corresponds to a decomposition of the computational domain into non-overlapping subdomains also called patches in the geometrical framework. We will present discontinuous Galerkin (dG) methods that allow for discontinuities across the subdomain (patch) boundaries. The required interface conditions are weakly imposed by the dG terms associated with the boundary of the subdomains. The construction and the corresponding discretization error analysis of such dG multi-patch IgA schemes is given for heterogeneous diffusion model problems in volumetric 2d and 3d domains as well as on open and closed surfaces. The theoretical results are confirmed by numerous numerical experiments which have been performed in G+SMO. The concept and the main features of the IgA library G+SMO are also described.
In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) t...
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ISBN:
(纸本)9783319198002;9783319197999
In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree.
In this work we discuss a newly developed class of robust and high-order accurate upwind schemes for wave equations in second-order form on curvilinear and overlapping grids. The schemes are based on embedding d'A...
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ISBN:
(纸本)9783319198002;9783319197999
In this work we discuss a newly developed class of robust and high-order accurate upwind schemes for wave equations in second-order form on curvilinear and overlapping grids. The schemes are based on embedding d'Alembert's exact solution for a local Riemann-type problem directly into the discretization (Banks and Henshaw, J Comput Phys 231(17):5854-5889, 2012). High-order accuracy is obtained using a single-step space-time scheme. Overlapping grids are used to represent geometric complexity. The method of manufactured solutions is used to demonstrate that the dissipation introduced through upwinding is sufficient to stabilize the wave equation in the presence of overlapping grid interpolation.
In this work we compare different families of nested quadrature points, i.e. the classic Clenshaw-Curtis and various kinds of Leja points, in the context of the quasi-optimal sparse grid approximation of random ellipt...
ISBN:
(纸本)9783319198002;9783319197999
In this work we compare different families of nested quadrature points, i.e. the classic Clenshaw-Curtis and various kinds of Leja points, in the context of the quasi-optimal sparse grid approximation of random elliptic PDEs. Numerical evidence suggests that both families perform comparably within such framework.
Recent developments in the SBP-SAT method have made available high-order interpolation operators (Mattsson and Carpenter, SIAM J Sci Comput 32(4):2298-2320, 2010). Such operators allow the coupling of different SBP me...
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ISBN:
(纸本)9783319198002;9783319197999
Recent developments in the SBP-SAT method have made available high-order interpolation operators (Mattsson and Carpenter, SIAM J Sci Comput 32(4):2298-2320, 2010). Such operators allow the coupling of different SBP methods across nonconforming interfaces of multiblock grids while retaining the three fundamental properties of the SBP-SAT method: strict stability, accuracy, and conservation. As these interpolation operators allow a more flexible computational mesh, they are appealing for complex geometries. Moreover, they are well suited for problems involving sliding meshes, like rotor/ stator interactions, wind turbines, helicopters, and turbomachinery simulations in general, since sliding interfaces are (almost) always nonconforming. With such applications in mind, this paper presents an accuracy analysis of these interpolation operators when applied to fluid dynamics problems on moving grids. The classical problem of an inviscid vortex transported by a uniform flow is analyzed: the flow is governed by the unsteady Euler equations and the vortex crosses a sliding interface. Furthermore, preliminary studies on a rotor/stator interaction are also presented.
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