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.     

An asymptotic finite element method for improvement of solutions of boundary layer problems

Summary A scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented. The proposed scheme improves the error bounds of the standard Galerkin finite element scheme by a factor of O(ⁿ⁺¹) (where is the small parameter andn is the order of the asymptotic approximation). Numerical results for linear second order O.D.E.'s are given and are compared with several other schemes.     

The formation of propylene fumarate oligomers for use in bioerodible bone cement composites

Poly(propylene fumarate) (PPF) oligomers were synthesized by step polymerization using bis(2-hydroxypropyl fumarate) or propylene bis(hydrogen maleate) as starting materials. Oligomers possessing identical degrees of polymerization (DP), but varying in their end group character (either hydroxyl or carboxyl) were first prepared and characterized, then used as part of a bone cement preparation consisting of oligomer, tricalcium phosphate, calcium carbonate, and methyl methacrylate. Compressive strength of the resulting composite appeared to be dependent on both the degree of polymerization of the PPF, and the nature of the oligomers' end groups.     

Novel continuum modeling of crystal surface evolution

We propose a novel approach to continuum modeling of the dynamics of crystal surfaces. Our model follows the evolution of an ensemble of step configurations, which are consistent with the macroscopic surface profile. Contrary to the usual approach where the continuum limit is achieved when typical surface features consist of many steps, our continuum limit is approached when the number of step configurations of the ensemble is very large. The model can handle singular surface structures such as corners and facets. It has a clear computational advantage over discrete models.     

On parallel asynchronous high-order solutions of parabolic PDEs

In achieving significant speed-up on parallel machines, a major obstacle is the overhead associated with synchronizing the concurrent processes. This paper presents high-orderparallel asynchronous schemes, which are schemes that are specifically designed to minimize the associated synchronization overhead of a parallel machine in solving parabolic PDEs. They are asynchronous in the sense that each processor is allowed to advance at its own speed. Thus, these schemes are suitable for single (or multi) user shared memory or (message passing) MIMD multiprocessors. Our approach is demonstrated for the solution of the multidimensional heat equation, of which we present a spatial second-order Parametric Asynchronous Finite-Difference (PAFD) scheme. The well-known synchronous schemes are obtained as its special cases. This is a generalization and expansion of the results in [5] and [7]. The consistency, stability and convergence of this scheme are investigated in detail. Numerical tests show that although PAFD provides the desired order of accuracy, its efficiency is inadequate when performed on each grid point.In an alternative approach that uses domain decomposition, the problem domain is divided among the processors. Each processor computes its subdomain mostly independently, while the PAFD scheme provides the solutions at the subdomains' boundaries. We use high-order finite-difference implicit scheme within each subdomain and determine the values at subdomains' boundaries by the PAFD scheme. Moreover, in order to allow larger time-step, we use remote neighbors' values rather than those of the immediate neighbors. Numerical tests show that this approach provides high efficiency and in the case which uses remote neighbors' values an almost linear speedup is achieved. Schemes similar to the PAFD can be developed for other types of equations [3].This research was supported by the fund for promotion of research at the Technion.     

Numerical calculation of transitional boundary layers

A computational procedure for compressible axisymmetric boundary layers, on bodies of revolution, in transition from laminar to turbulent flow, is introduced. The procedure is an extension of a former method, due to Patankar and Spalding. The flow field is computed by solution of four simultaneous equations for the momentum, the thermal energy, the turbulence energy amplitude and the turbulent scale. The results show good agreement with existing theoretical and experimental data.     

Spinup from Rest of Immiscible Fluids

We investigate the spinup from rest of two immiscible fluids of different densities and kinematic viscosities in a vertically mounted cylinder. Our attention is restricted to small internal rotational Froude numbers, in which case the interface remains essentially horizontal. By requiring small enough Ekman numbers, Wedemeyer's (1964) approximation may be used to obtain partial differential equations describing the inward convection and diffusion of the azimuthal velocity in both layers. Solutions are found illustrating the effects of varying the density, viscosity, and height of each layer. Qualitative agreement with experiment is reported.     

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