In this work we deal with a different technique from the considered one in Clavero et al. (IMA J Numer Anal 26: 155-172, 2006;Appl NumerMath 27: 211231, 1998), to analyze the uniform convergence of some numerical meth...
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ISBN:
(纸本)9783319257273;9783319257259
In this work we deal with a different technique from the considered one in Clavero et al. (IMA J Numer Anal 26: 155-172, 2006;Appl NumerMath 27: 211231, 1998), to analyze the uniform convergence of some numerical methods which have been used to solve successfully two dimensional parabolic singularly perturbed problems of convection-diffusion type. For getting this, we split the discretization methods in a two stage procedure where, firstly, we semidiscretize in space, using the classical upwind scheme on a piecewise uniform Shishkin mesh, and, secondly, we integrate in time the Initial Value Problems derived from the first stage, by using the implicit Euler method. The analysis combines a suitable maximum semidiscrete principle joint to some well known techniques used in the proof of the uniform convergence of numerical schemes for elliptic singularly perturbed problems. We prove that the stiff initial value problems resulting of the spatial semidiscretization processes, have a unique solution which converges uniformly with respect to the singular perturbation parameter. Using this technique, some restrictions among the discretization parameters, which appeared in the uniform convergence analysis in Clavero et al. (Appl Numer Math 27:211-231, 1998), can be removed. Some numerical results corroborate in practice the robustness of the numerical method, according to the theoretical results.
The classical do-nothing condition is very often prescribed at outflow boundaries for fluid dynamical problems. However, it has a severe drawback in the context of the Navier-Stokes equations, because not even existen...
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ISBN:
(纸本)9783319257273;9783319257259
The classical do-nothing condition is very often prescribed at outflow boundaries for fluid dynamical problems. However, it has a severe drawback in the context of the Navier-Stokes equations, because not even existence of weak solutions can be shown. The reason is that this boundary condition does not exhibit any control about inflow across such boundaries. This has also severe impact onto the stability of numerical algorithms for flows at higher Reynolds number. A modification of this boundary condition is one possibility to circumvent these drawbacks. This paper addresses such modifications in the context of the skew-symmetric formulation of the convective term. Moreover, we introduce a parameter which gives the possibility to downsize possible inflow even more and to enhance the stability further. Numerical examples illustrate the effectiveness of the approach.
In the description of water waves, dispersion is one of the most important physical properties;it specifies the propagation speed as function of the wavelength. Accurate modelling of dispersion is essential to obtain ...
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ISBN:
(纸本)9783319198002;9783319197999
In the description of water waves, dispersion is one of the most important physical properties;it specifies the propagation speed as function of the wavelength. Accurate modelling of dispersion is essential to obtain high-quality wave propagation results. The relation between speed and wavelength is given by a non-algebraic relation;for finite element/difference methods this relation has to be approximated and leads to restrictions for waves that are propagated correctly. By using a spectral implementation dispersion can be dealt with exactly above flat bottom using a pseudo-differential operator so that all wavelengths can be propagated correctly. However, spectral methods are most commonly applied for problems in simple domains, while most water wave applications need complex geometries such as (harbour) walls, varying bathymetry, etc.;also breaking of waves requires a local procedure at the unknown position of breaking. This paper deals with such inhomogeneities in space;the models are formulated using Fourier integral operators and include non-trivial localization methods. The efficiency and accuracy of a so-called spatial-spectral implementation is illustrated here for a few test cases: wave run-up on a coast, wave reflection at a wall and the breaking of a focussing wave. These methods are included in HAWASSI software (Hamiltonian Wave-Ship-Structure Interaction) that has been developed over the past years.
We present a locally conservative spectral least-squares formulation for the scalar diffusion-reaction equation in curvilinear coordinates. Careful selection of a least squares functional and compatible finite dimensi...
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ISBN:
(纸本)9783319198002;9783319197999
We present a locally conservative spectral least-squares formulation for the scalar diffusion-reaction equation in curvilinear coordinates. Careful selection of a least squares functional and compatible finite dimensional subspaces for the solution space yields the conservation properties. Numerical examples confirm the theoretical properties of the method.
We present two-sided a posteriori error estimates for isogeometric discretization of elliptic problems. These estimates, derived on functional grounds, provide robust, guaranteed and sharp two-sided bounds of the exac...
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ISBN:
(数字)9783319233154
ISBN:
(纸本)9783319233154;9783319233147
We present two-sided a posteriori error estimates for isogeometric discretization of elliptic problems. These estimates, derived on functional grounds, provide robust, guaranteed and sharp two-sided bounds of the exact error in the energy norm. Moreover, since these estimates do not contain any unknown/generic constants, they are fully computable, and thus provide quantitative information on the error. The numerical realization and the quality of the computed error distribution are addressed. The potential of the proposed estimates are illustrated using several computational examples.
In this contribution we present a sub-cell discretization method for the computation of the interface velocities involved in the convective terms of the incompressible Navier-Stokes equations. We compute an interface ...
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ISBN:
(纸本)9783319198002;9783319197999
In this contribution we present a sub-cell discretization method for the computation of the interface velocities involved in the convective terms of the incompressible Navier-Stokes equations. We compute an interface velocity by solving a local two-point boundary value problem (BVP) iteratively. To account for the two-dimensionality of the interface velocity we introduce a constant cross-flux term in our computation. The discretization scheme is used to simulate the flow in a lid-driven cavity.
We outline and extend results for an explicit local time stepping (LTS) strategy designed to operate with the discontinuous Galerkin spectral element method (DGSEM). The LTS procedure is derived from Adams-Bashforth m...
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ISBN:
(纸本)9783319198002;9783319197999
We outline and extend results for an explicit local time stepping (LTS) strategy designed to operate with the discontinuous Galerkin spectral element method (DGSEM). The LTS procedure is derived from Adams-Bashforth multirate time integration methods. The new results of the LTS method focus on parallelization and reformulation of the LTS integrator to maintain conservation. Discussion is focused on a moving mesh implementation, but the procedures remain applicable to static meshes. In numerical tests, we demonstrate the strong scaling of a parallel, LTS implementation and compare the scaling properties to a parallel, global time stepping (GTS) Runge-Kutta implementation. We also present time-step refinement studies to show that the redesigned, conservative LTS approximations are spectrally accurate in space and have design temporal accuracy.
Strong stability preserving (SSP) high order time discretizations were developed to address the need for nonlinear stability properties in the numerical solution of hyperbolic partial differential equations with disco...
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ISBN:
(纸本)9783319198002;9783319197999
Strong stability preserving (SSP) high order time discretizations were developed to address the need for nonlinear stability properties in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. These methods preserve the monotonicity properties (in any norm, seminorm or convex functional) of the spatial discretization coupled with first order Euler time stepping. This review paper describes the state of the art in SSP methods.
In this paper, a boundary value problem for a system of two singularly perturbed second order delay differential equations is considered on the interval [0, 2]. The components of the solution of this system exhibit bo...
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ISBN:
(纸本)9783319257273;9783319257259
In this paper, a boundary value problem for a system of two singularly perturbed second order delay differential equations is considered on the interval [0, 2]. The components of the solution of this system exhibit boundary layers at x = 0 and x = 2 and interior layers at x D 1. A numerical method composed of a classical finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical illustration provided support the theory.
We present an algorithm that reformulates existing methods to construct higher-order mimetic differential operators. Constrained linear optimization is the key idea of this resulting algorithm. The authors exemplified...
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ISBN:
(纸本)9783319198002;9783319197999
We present an algorithm that reformulates existing methods to construct higher-order mimetic differential operators. Constrained linear optimization is the key idea of this resulting algorithm. The authors exemplified this algorithm by constructing an eight-order-accurate one-dimensional mimetic divergence operator. The algorithm computes the weights that impose the mimetic condition on the constructed operator. However, for higher orders, the computation of valid weights can only be achieved through this new algorithm. Specifically, we provide insights on the computational implementation of the proposed algorithm, and some results of its application in different test cases. Results show that for all of the proposed test cases, the proposed algorithm effectively solves the problem of computing valid weights, thus constructing higher-order mimetic operators.
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