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Finite-difference Methods (finite-difference + methods)
Selected AbstractsStable high-order finite-difference methods based on non-uniform grid point distributionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2008Miguel Hermanns Abstract It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundström theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q,N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q=N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2005K. S. Erduran Abstract A hybrid scheme composed of finite-volume and finite-difference methods is introduced for the solution of the Boussinesq equations. While the finite-volume method with a Riemann solver is applied to the conservative part of the equations, the higher-order Boussinesq terms are discretized using the finite-difference scheme. Fourth-order accuracy in space for the finite-volume solution is achieved using the MUSCL-TVD scheme. Within this, four limiters have been tested, of which van-Leer limiter is found to be the most suitable. The Adams,Basforth third-order predictor and Adams,Moulton fourth-order corrector methods are used to obtain fourth-order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ,HYWAVE', based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi-chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd. [source] Modelling of lossy curved surfaces in the 3-D frequency-domain finite-difference methodsINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 5 2006Riku M. Mäkinen Abstract A conformal first-order or Leontovic surface-impedance boundary condition (SIBC) for the modelling of fully three-dimensional (3-D) lossy curved surfaces in a Cartesian grid is presented for the frequency-domain finite-difference (FD) methods. The impedance boundary condition is applied to auxiliary tangential electric and magnetic field components defined at the curved surface. The auxiliary components are subsequently eliminated from the formulation resulting in a modification of the local permeability value at boundary cells, allowing the curved 3-D surface to be described in terms of Cartesian grid components. The proposed formulation can be applied to model skin-effect loss in time-harmonic driven problems. In addition, the impedance matrix can be used as a post-processor for the eigenmode solver to calculate the wall loss. The validity of the proposed model is evaluated by investigating the quality factors of cylindrical and spherical cavity resonators. The results are compared with analytic solutions and numerical reference data calculated with the commercial software package CST Microwave StudioÔ (MWS). The convergence rate of the results is shown to be of second-order for smooth curved metal surfaces. The overall accuracy of the approach is comparable to that of CST MWSÔ. Copyright © 2006 John Wiley & Sons, Ltd. [source]
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