ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Cover image for Vol. 93 Issue 6‐7

Special Issue: Partial Differential Equations: Theory, Applications, Simulations

June 2013

Volume 93, Issue 6-7

Pages 367–498

Issue edited by: Raimund Bürger, Ernst P. Stephan, Olaf Steinbach

  1. Cover Picture

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. You have free access to this content
      Cover Picture: ZAMM 6–7 / 2013

      Version of Record online: 4 JUN 2013 | DOI: 10.1002/zamm.201390011

      See in this issue, pages 387–402: Triangulation at time t = 0 with 45856 elements used for the computation of hypersonic flow and its detail near the airfoil; Mach number distribution at time instants t = 0.00058 s for the far-field velocity 680 m/s M = 2.0) and Reynolds numbers Re = 107

  2. Issue Information

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. You have free access to this content
      Issue Information: ZAMM 6–7 / 2013

      Version of Record online: 4 JUN 2013 | DOI: 10.1002/zamm.201390012

  3. Contents

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. You have free access to this content
      Contents: ZAMM 6–7 / 2013 (pages 367–371)

      Version of Record online: 4 JUN 2013 | DOI: 10.1002/zamm.201309306

  4. Editorial

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. You have free access to this content
      Editorial: ZAMM 6–7 / 2013 (page 372)

      Raimund Bürger, Olaf Steinbach and Ernst Stephan

      Version of Record online: 4 JUN 2013 | DOI: 10.1002/zamm.201309366

  5. Original Papers

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. Spectral WENO schemes with Adaptive Mesh Refinement for models of polydisperse sedimentation (pages 373–386)

      R. Bürger, P. Mulet and L.M. Villada

      Version of Record online: 11 JUN 2012 | DOI: 10.1002/zamm.201100189

      Thumbnail image of graphical abstract

      The sedimentation of a polydisperse suspension with particles belonging to N size classes (species) can be described by a system of N nonlinear, strongly coupled scalar first-order conservation laws. Its solutions usually exhibit kinematic shocks separating areas of different composition. Based on the so-called secular equation [J. Anderson, Lin. Alg. Appl. 246, 49–70 (1996)], which provides access to the spectral decomposition of the Jacobian of the flux vector for this class of models, Bürger et al. [J. Comput. Phys. 230, 2322–2344 (2011)] proposed a spectral weighted essentially non-oscillatory (WENO) scheme for the numerical solution of the model. It is demonstrated that the efficiency of this scheme can be improved by the technique of Adaptive Mesh Refinement (AMR), which concentrates computational effort on zones of strong variation. Numerical experiments for the cases N = 4 and N = 7 are presented.

    2. DGFEM for the analysis of airfoil vibrations induced by compressible flow (pages 387–402)

      J. Česenek, M. Feistauer and A. Kosík

      Version of Record online: 20 FEB 2013 | DOI: 10.1002/zamm.201100184

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      The subject of the paper is the numerical simulation of the interaction of two-dimensional compressible viscous flow and a vibrating airfoil. A solid airfoil with two degrees of freedom performs rotation around an elastic axis and oscillations in the vertical direction. The numerical simulation consists of the solution of the Navier-Stokes system by the discontinuous Galerkin method coupled with a system of nonlinear ordinary differential equations describing the airfoil motion. The time-dependent computational domain and a moving grid are taken into account by the arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations. The developed method is robust with respect to the magnitude of the Mach number and Reynolds number. Its applicability is demonstrated by numerical experiments.

    3. Sparse space-time Galerkin BEM for the nonstationary heat equation (pages 403–413)

      A. Chernov and Ch. Schwab

      Version of Record online: 29 MAY 2012 | DOI: 10.1002/zamm.201100192

      We construct and analyze sparse tensorized space-time Galerkin discretizations for boundary integral equations resulting from the boundary reduction of nonstationary diffusion equations with either Dirichlet or Neumann boundary conditions. The approach is based on biorthogonal multilevel subspace decompositions and a weighted sparse tensor product construction. We compare the convergence behavior of the proposed method to the standard full tensor product discretizations. In particular, we show for the problem of nonstationary heat conduction in a bounded two- or three-dimensional spatial domain that low order sparse space-time Galerkin schemes are competitive with high order full tensor product discretizations in terms of the asymptotic convergence rate of the Galerkin error in the energy norms, under lower regularity requirements on the solution.

  6. Book Review

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
  7. Original Papers

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. On the positivity of solutions of systems of stochastic PDEs (pages 414–422)

      J. Cresson, M. Efendiev and Stefanie Sonner

      Version of Record online: 6 AUG 2012 | DOI: 10.1002/zamm.201100167

      We study the positivity of solutions of a class of semi-linear parabolic systems of stochastic partial differential equations by considering random approximations. For the family of random approximations we derive explicit necessary and sufficient conditions such that the solutions preserve positivity. These conditions imply the positivity of the solutions of the stochastic system for both Itô's and Stratonovich's interpretation of stochastic differential equations.

    2. High-order finite volume schemes based on defect corrections (pages 423–436)

      A. Filimon, M. Dumbser and C.-D. Munz

      Version of Record online: 24 SEP 2012 | DOI: 10.1002/zamm.201200007

      Thumbnail image of graphical abstract

      For the approximation of steady state solutions, we propose an iterated defect correction approach to achieve higher-order accuracy. The procedure starts with the steady state solution of a low-order scheme, in general a second order one. The higher-order reconstruction step is applied a posteriori to estimate the local discretization error of the lower-order finite volume scheme. The defect is then used to iteratively shift the basic lower-order scheme to the desired higher-order accuracy given by the polynomial reconstruction. Hence, instead of solving the high-order discrete equations the loworder basic scheme is solved several times. This avoids that the high-order reconstruction with a large stencil has to be implemented into an existing basic solver and can be seen as a non-intrusive approach to higher-order accuracy.

    3. On the dual-mixed formulation for an exterior Stokes problem (pages 437–445)

      G.N. Gatica, G.C. Hsiao, S. Meddahi and F.-J. Sayas

      Version of Record online: 17 DEC 2012 | DOI: 10.1002/zamm.201200001

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      This paper is concerned with a dual-mixed formulation for a three dimensional exterior Stokes problem via boundary integral equation methods. Here velocity, pressure and stress are the main unknowns. Following a similar analysis given recently for the Laplacian, we are able to extend the classical Johnson & Nédélec procedure to the present case, without assuming any restrictive smoothness requirement on the coupling boundary, but only Lipschitz-continuity. More precisely, after using the incompressibility condition to eliminate the pressure, we consider the resulting velocity-stress approach with a Neumann boundary condition on an annular bounded domain, and couple the underlying equations with only one boundary integral equation arising from the application of the normal trace to the Green representation formula in the exterior unbounded region. As a result, we obtain a saddle point operator equation, which is then analyzed by the wellknown Babuška-Brezzi theory: in particular, the well-posedness of the formulation will be established.

  8. Editor's Choice

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. You have free access to this content
      Dirichlet-transmission problems for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in Riemannian manifolds (pages 446–458)

      M. Kohr, C. Pintea and W.L. Wendland

      Version of Record online: 7 AUG 2012 | DOI: 10.1002/zamm.201100194

      Thumbnail image of graphical abstract

      In this paper we use a layer potential analysis to show the existence of solutions for a Dirichlet-transmission problem for pseudodifferential Brinkman operators on Sobolev and Besov spaces associated to Lipschitz domains in compact boundaryless Riemannian manifolds. Compactness and invertibility properties of corresponding layer potential operators on Lp, Sobolev or Besov scales are also obtained.

  9. Original Papers

    1. Top of page
    2. Cover Picture
    3. Issue Information
    4. Contents
    5. Editorial
    6. Original Papers
    7. Book Review
    8. Original Papers
    9. Editor's Choice
    10. Original Papers
    1. An extended Discontinuous Galerkin and Spectral Difference Method with modal filtering (pages 459–464)

      A. Meister, S. Ortleb, T. Sonar and M. Wirz

      Version of Record online: 6 AUG 2012 | DOI: 10.1002/zamm.201200051

      Thumbnail image of graphical abstract

      We give a short overview of an extended Discontinuous Galerkin and Spectral Difference Method using PKD polynomials on triangular grids. A corresponding exponential filter is used to avoid oscillations near discontinuous solutions and to give some stabilization for nonsmooth testcases.

    2. Criteria for nonuniqueness of Riemann solutions to compressible duct flows (pages 465–475)

      E. Han, M. Hantke and G. Warnecke

      Version of Record online: 17 DEC 2012 | DOI: 10.1002/zamm.201100176

      Thumbnail image of graphical abstract

      The Riemann solutions without vacuum states for compressible duct flows have been completely constructed in E. Han, M. Hantke, and G. Warnecke, J. Hyp. Differ. Equ. 9, 403–449 (2012). However, the nonuniqueness of Riemann solutions due to a bifurcation of wave curves in state space is still an open problem. The purpose of this paper is to single out the physically relevant solution among all the possible Riemann solutions by comparing them with the numerical results of the axisymmetric Euler equations. Andrianov and Warnecke [SIAM J. Appl. Math. 64, 78–901 (2004)] suggested using the entropy rate admissibility criterion to rule out the unphysical solutions. However, this criterion is not true for some test cases, i.e. the numerical result for axisymmetric three dimensional flows picks up an exact solution which does not satisfy the entropy rate admissibility criterion. Moreover, numerous numerical experiments show that the physically relevant solution is always located on a certain branch of the L–M curves.

    3. Is the one-equation coupling of finite and boundary element methods always stable? (pages 476–484)

      G. Of and O. Steinbach

      Version of Record online: 11 MAR 2013 | DOI: 10.1002/zamm.201100188

      In this paper we present a sufficient and necessary condition to ensure the ellipticity of the bilinear form which is related to the one-equation coupling of finite and boundary element methods to solve a scalar free space transmission problem for a second order uniform elliptic partial differential equation in the case of general Lipschitz interfaces. This condition relates the minimal eigenvalue of the coefficient matrix in the bounded interior domain to the contraction constant of the shifted double layer integral operator which reflects the shape of the interface. This paper extends and improves earlier results [O. Steinbach, SIAM J. Numer. Anal. 49, 1521–1531 (2011)] on sufficient conditions, but now includes also necessary conditions. Numerical examples confirm the theoretical results on the sharpeness of the presented estimates.

    4. Diffraction from a three-quarter-plane using an abstract Babinet principle (pages 485–491)

      F.-O. Speck

      Version of Record online: 19 APR 2012 | DOI: 10.1002/zamm.201100175

      The modelling of diffraction of time-harmonic electromagnetic or acoustic waves from obstacles and screens leads to boundary value problems for the three-dimensional Helmholtz equation with Dirichlet, Neumann, or other conditions on the boundary. A prominent example is the problem of diffraction from a quarter-plane in ℝ3, which admits an explicit solution. In this paper the Dirichlet and Neumann problems for the three-quarter-plane are solved by an algebraic trick: the matricial coupling of operators associated to “dual” boundary value problems, a kind of abstract Babinet principle.

    5. High precision modeling towards the 10-20 level (pages 492–498)

      E.P. Stephan, M. Andres, L. Banz, A. Costea, L. Nesemann, C. Lämmerzahl, E. Hackmann, S. Herrmann and B. Rievers

      Version of Record online: 21 MAY 2013 | DOI: 10.1002/zamm.201200074

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      The requirements for accurate numerical simulations are increasing steadily. Modern high precision physics experiments now exceed the achievable numerical accuracy of standard commercial and scientific simulation tools. One example are optical resonators for which changes in the optical length are now commonly measured to 10-15 precision. The achievable measurement accuracy for resonators and cavities is directly influenced by changes in the distances between the optical components. If deformations in the range of 10-15 occur, those effects cannot be modeled and analyzed anymore with standard methods based on double precision data types. New experimental approaches point out that the achievable experimental accuracies may improve up to the level of 10-17 in the near future. For the development and improvement of high precision resonators and the analysis of experimental data, new methods have to be developed which enable the needed level of simulation accuracy. Therefore we plan the development of new high precision algorithms for the simulation and modeling of thermo-mechanical effects with an achievable accuracy of 10-20. In this paper we analyze a test case and identify the problems on the way to this goal.

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