On a fast direct elliptic solver by a modified Fourier method

We describe high order numerical algorithms for the solution of second order elliptic equations in rectangular domains. These algorithms are based on the Fourier method in combination with a subtraction procedure. The singularities at the corner points, arising due to non-smoothness of the boundaries, are treated explicitly using properly constructed singular corner functions. The present algorithm is a generalization of the Fast Poisson Solver developed in our previous paper. This revised version was published online in August 2006 with corrections to the Cover Date.     

Convection currents in a vertical layer with a space-modulated temperature distribution on the boundaries

Steady convective motions in a plane vertical fluid layer are investigated. The temperature along the boundaries of the layer varies harmonically and has different average values on each of the boundaries. Thus space-period modulation of the temperature of the walls is assigned along with average lateral heating of the layer. The form of the plane steady motions and regions of existence of through currents and currents of cellular structure are found for various values of the parameters of the problem by the finite difference grid-point method. The dependence of the main characteristics of fluid motion on the Grashof number is determined. The results presented in the article pertain to the case when the period of modulation of the temperature of the boundaries coincides with the wavelength of the critical mode of a plane-parallel current. A numerical investigation of supercritical motions in a vertical layer with plane isothermal boundaries heated to a different temperature was carried out in [1–3]. The effect of a space-periodic inhomogeneity due to curvature of walls on the form and stability of convective motions in a vertical layer with lateral heating was examined in [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–25, September–October, 1978.The author thanks E. M. Zhukhovitskii for formulating the problem and supervising the work and G. Z. Gershuni for discussions and useful comments.     

Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.     

A parallel spectral Fourier–Nonlinear Galerkin algorithm for simulation of turbulence

We present a high-order parallel algorithm, which requires only the minimum interprocessor communication dictated by the physical nature of the problem at hand. The parallelization is achieved by domain decomposition. The discretization in space is performed using the Local Fourier Basis method. The continuity conditions on the interfaces are enforced by adding homogeneous solutions. Such solutions often have fast decay properties, which can be utilized to minimize interprocessor communication. In effect, the predominant part of the computation is performed independently in the subdomains (processors) or using only local communication. A novel element of the present parallel algorithm is the incorporation of a Nonlinear Galerkin strategy to accelerate the computation and stabilize the time integration process. The basic idea of this approach consists of decomposition of the variables into large scale and small scale components with different treatment of these large and small scales. The combination of the Multidomain Fourier techniques with the Nonlinear Galerkin (NLG) algorithm is applied here to solve incompressible Navier–Stokes equations. Results are presented on direct numerical simulation of two-dimensional homogeneous turbulence using the NLG method. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 699–715, 1997     

On the stability of mixed convective motions in a vertical layer with wave-like boundaries

The stability of the stationary and oscillatory convective motions which develop in a vertical layer with periodically curved boundaries is studied for the case of longitudinal fluid injection. The amplitude of the boundary undulations and the flow of fluid along the layer are both assumed to be small, and methods of perturbation theory are used. The characteristic properties of the incremental spectrum of the spatially periodic motions are studied and the most dangerous types of perturbations as well as the forms of the stability regions are determined.

Theoretical investigations of the effect of spatial inhomogeneity of the boundary conditions on the stability of convection were sparse, and they deal mainly with horizontal layers of fluid /1–3/. Stationary, spatially periodic motions in a vertical layer with curved boundaries were investigated in /4/ for the case of free convection (when the flow was closed), and their stability was investigated in /5/. It was established that the presence of a small but finite flow of fluid along the layer leads to an increase in the number of different modes of flow, and to the appearance of non-stationary convective motions in the region near the threshold.     

On the stability of space-periodic convective flows in a vertical layer with sinuous boundaries : PMM vol. 43, no. 6, 1979, pp. 998–1007

The problem of convection in a vertical layer with harmonically distorted boundaries is examined by perturbation theory methods for a small amplitude of sinuosity. The solutions obtained are applicable both in the stability region as well as in the supercritical region of the plane-parallel flow. The stability of the solutions found is investigated with respect to a certain class of space-bounded perturbations that are not necessarily space-periodic. The method of amplitude functions [1], generalized to the case of curved boundaries, is used. The Grashof critical number is found as a function of the period of sinuosity and the form of the neutral curve for the space-periodic motions and their stability region are obtained. It is established that if the deformation period of the boundaries is close to the wavelength of the critical perturbation for the plane-parallel flow or is twice as great, then as the Grashof number grows stability loss does not occur and the motion's amplitude changes continuously (cf. [2 — 4]). A comparison is made with the results of the numerical calculation in [5], An attempt was made in [6] to construct a stationary periodic motion in a layer with weakly-deformed boundaries, in the form of series in powers of a small sinuosity amplitude. However, the solution obtained diverges in a neighborhood of the neutral curve of the plane-parallel flow and approximates unstable motion in the supercritical region of the unperturbed problem. Flows under a finite sinuosity amplitude are calculated by the net method in [5] wherein the stability of the flows was investigated as well, but only with respect to perturbations with wave numbers that are multiples of 2π/l, where l is the length of the calculated region.     

Finite-amplitude regimes of mixed convection in a vertical layer with undulating boundaries

The method of finite differences is used to construct convective motions in a vertical layer with sinusoidally curved boundaries, fluid being pumped through longitudinally. Apart from steady and oscillation regimes, found earlier by analytical means for small amplitudes of undulation and slow pumping through [1, 2], new, essentially nonlinear, types of motion are discovered in the form of two-stroke cycles, and also of complex multi-revolution cycles which are two-dimensional resonance tori. The regions are determined in which regimes of various types exist.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–20, January–February, 1987.The author is grateful to E. M. Zhukhovitskii for constant interest in the study, and also to V. S. Anishchenko and A. A. Nepomnyashchii for useful discussions.