A unified solution framework is presented for one-, two- or three-dimensional complex non-symmetric eigenvalue problems, respectively governing linear modal instability of incompressible fluid flows in rectangular domains having two, one or no homogeneous spatial directions. The solution algorithm is based on subspace iteration in which the spatial discretization matrix is formed, stored and inverted serially. Results delivered by spectral collocation based on the Chebyshev-Gauss-Lobatto (CGL) points and a suite of high-order finite-difference methods comprising the previously employed for this type of work Dispersion-Relation-Preserving (DRP) and Padé finite-difference schemes, as well as the Summationby- parts (SBP) and the new high-order finite-difference scheme of order q (FD-q) have been compared from the point of view of accuracy and efficiency in standard validation cases of temporal local and BiGlobal linear instability. The FD-q method has been found to significantly outperform all other finite difference schemes in solving classic linear local, BiGlobal, and TriGlobal eigenvalue problems, as regards both memory and CPU time requirements. Results shown in the present study disprove the paradigm that spectral methods are superior to finite difference methods in terms of computational cost, at equal accuracy, FD-q spatial discretization delivering a speedup of ð (10 4). Consequently, accurate solutions of the three-dimensional (TriGlobal) eigenvalue problems may be solved on typical desktop computers with modest computational effort

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<N through the use of the proposed non‐uniform grids. Discussions are complemented by spectra and pseudospectra of the finite‐difference operators. Copyright © 2007 John Wiley & Sons, Ltd.

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude\ud perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two(and periodic in one)or all three spatial directions.After a brief exposition of the theory,some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton–Krylov time-stepping method into a standard\ud finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in\ud flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the\ud spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability\ud equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse\ud matrix treatment, all these problems may now be solved on standard desktop computer

Theoretical models for the thermal response of vertical geothermal boreholes often assume that the characteristic time of variation of the heat injection rate is much larger than the characteristic diffusion time across the borehole. In this case, heat transfer inside the borehole and in its immediate surroundings is quasi-steady in the first approximation, while unsteady effects enter only in the far field. Previous studies have exploited this disparity of time scales, incorporating approximate matching conditions to couple the near-borehole region with the outer unsteady temperature field. In the present work matched asymptotic expansion techniques are used to analyze the heat transfer problem, delivering a rigorous derivation of the true matching condition between the two regions and of the correct definition of the network of thermal resistances that represents the quasi-steady solution near the borehole. Additionally, an apparent temperature due to the unsteady far field is identified that needs to be taken into account by the near-borehole region for the correct computation of the heat injection rate. This temperature differs from the usual mean borehole temperature employed in the literature.