L. Vozovoi

A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

In this paper we propose a direct method for the solution of the Poisson equation in rectangular regions. It has an arbitrary order accuracy and low CPU requirements which makes it practical for treating large-scale problems. The method is based on a pseudospectral Fourier approximation and a polynomial subtraction technique. Fast convergence of the Fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions. These functions have the same discontinuities at the corner points as the sought solution. In addition to this, they satisfy the Laplace equation so that the subtraction procedure does not generate nonperiodic, nonhomogeneous terms. The solution of a boundary value problem is obtained in a series form in O(N log N) floating point operations, where N2 is the number of grid nodes. Evaluating the solution at all N2 interior points requires O(N2 log N) operations.

Spectral multidomain technique with local Fourier basis

A novel domain decomposition method for spectrally accurate solutions of PDEs is presented. A Local Fourier Basis technique is adapted for the construction of the elemental solutions in subdomains.C1 continuity is achieved on the interfaces by a matching procedure using the analytical homogeneous solutions of a one dimensional equation. The method can be applied to the solution of elliptic problems of the Poisson or Helmholtz type as well as to time discretized parabolic problems in one or more dimensions. The accuracy is tested for several stiff problems with steep solutions.The present domain decomposition approach is particularly suitable for parallel implementations, in particular, on MIMD type parallel machines.

Analysis and Application of Fourier--Gegenbauer Method to Stiff Differential Equations

The Fourier--Gegenbauer (FG) method, introduced by [Gottlieb, Shu, Solomonoff, and Vandeven, ICASE Report 92-4, Hampton, VA, 1992] is aimed at removing the Gibbs phenomenon; that is, recovering the point values of a nonperiodic function from its Fourier coefficients. In this paper, we discuss some numerical aspects of the FG method related to its {\em pseudospectral\/} implementation. In particular, we analyze the behavior of the Gegenbauer series with a moderate (several hundred) number of terms suitable for computations. We also demonstrate the ability of the FG method to get a spectrally accurate approximation on small subintervals for rapidly oscillating functions or functions having steep profiles. Bearing on the previous analysis, we suggest a high-order spectral Fourier method for the solution of nonperiodic differential equations. It includes a polynomial subtraction technique to accelerate the convergence of the Fourier series and the FG algorithm to evaluate derivatives on the boundaries of nonperiodic functions. The present hybrid Fourier--Gegenbauer (HFG) method possesses better resolution properties than the original FG method. The precision of this method is demonstrated by solving stiff elliptic problems with steep solutions.

Spectral multidomain technique with local Fourier basis II: decomposition into cells

The spectral multidomain method for the solution of 2-D elliptic and parabolic PDEs is developed. The computational region is decomposed into rectangular cells. A Local Fourier Basis technique is implemented for the discretization in space. Such a technique enables the global (typically ∼104–105) matching relations for the interface unknows to be decoupled into a set of relations for only few interface points at a time.

Multidomain local Fourier method for PDEs in complex geometries

Abstract A low communication parallel algorithm is developed for the solution of time-dependent nonlinear PDEs. The parallelization is achieved by domain decomposition. The discretization in time is performed via a third-order semi-implicit stiffly stable scheme. The elemental solutions in the subdomains are constructed using a high-order method with the local Fourier basis (LFB). The continuity of the global solution is accomplished by a point-wise matching of the local subsolutions on the interfaces. The matching relations are derived in terms of the jumps on the interfaces. The LFB method enables splitting a two-dimensional problem with global coupling of the interface unknowns into a set of uncoupled one-dimensional differential equations. Localization properties of an elliptic operator, resulting from the discretization in time of a time-dependent problem, are utilized in order to simplify the matching relations. In effect, only local (neighbor-to-neighbor) communication between the processors becomes necessary. The present method allows the treatment of problems in various complex geometries by the mapping of curvilinear domains into simpler (rectangular or circular) regions with subsequent matching of local solutions. The operator with nonconstant coefficients, obtained in the transformed domain, is preconditioned by an appropriate constant coefficient operator, easily inverted by the LFB. The problem is then solved with spectral accuracy by (a rapidly convergent) conjugate gradient iteration.

Two-dimensional parallel solver for the solution of Navier-Stokes equations with constant and variable coefficients using ADI on cells

Abstract The paper proposed a new algorithm for the parallel solution of two-dimensional Navier–Stokes type equation with constant and non-constant coefficients which is mapped onto cell topology. This paper is a further development in the application of the local Fourier methods to the solutions of PDEs in multidomain regions. The extension of the above solution to problems with non-constant coefficients is suggested via spectral multidomain preconditioner. This approach is efficient when we have good local approximations in each subdomain. By dividing the computational domain into a large enough number of subdomains we can guarantee it. The new achievement here is that we are able to handle decomposition of the domain into cells that is the decomposed in both directions, x and y . An appropriate alternate direction implicit (ADI) scheme was developed. It enables the reduction of a 2-D problem to a collection of uncoupled 1-D ODEs. In effect, the 1-D solver becomes the basic routine to solve a 2-D problem using splitting of the differential operators by ADI. Detailed performance analysis is given where the issue of the communication among the domains (processors) is examined. We show that by using the Richardson method only local communication is required. The algorithm was implemented on IBM SP2, network of ALPHA workstations, and MOSIX [A. Barak, S. Guday, R. Wheeler, The MOSIX Distributed Operating System, Load Balancing for UNIX, Lecture Notes in Computer Science, Vol. 672, Springer-Verlag, 1993; A. Barak, O. Laden, Z. Yarom, The NOW MOSIX and its Preemptive Process Migration Scheme, IEEE TCOS 7 (2) (1995) 5–11] which is a network of i586. All are implemented using the PVM software package and the same ADI program was running on these different multiprocessor configurations. It achieved efficiency of 55–70% depending on the multiprocessor.

Parallel implementation of non-linear evolution problems using parabolic domain decomposition

Abstract We present implementation of parallel algorithms for the numerical solution of nonlinear time-dependent partial differential equations of parabolic type arising from complex large scale problems. The parallelization is achieved by using domain decomposition (DD) techniques. The essential feature of this algorithm is that the spatial discretization in each subdomain is performed by using spectral method with the Local Fourier Basis (LFB) [1]. Our solutions are based on a special projection technique that employed to localize functions in a smooth way on the extended subdomain. The current paper continue the flow of our previous results [1,4,12,13] on spectral multidomain algorithm. The application of the Parabolic Domain Decomposition (PDD) approach along with the LFB is shown to be very efficient when applied to a 2-dimensional domain splitted into strips and rectangular cells. In this case, all matching relations become completely uncoupled (at the price of some overlapping of subdomains required by the LFB implementation). Thus, all communication is reduced to interactions between neighbouring elements and thus fits to scalable message-passing multiprocessor. The continuity of a global solution is attained by using a direct point-wise matching of the local subsolutions on the interfaces. The implementation of the LFB technique enables us to trade a 2-D problem with the overall coupling of the interface unknown into a set of uncoupled 1-D differential equations with simple matching relations. 2-D Navier-Stokes type modeling equation is implemented on the Meiko message-passing type scalable MIMD multiprocessor. Detailed performance analysis is presented. The algorithm is scalable in the sense that problems of equal size have the same speedup when the number of processors increase because the communication is reduced from global to local and, thus, the solution depends on direct neighboring processors. When the size if each sub-domain in each processor is large enough a linear speedup is achieved. The implementation strategy of the algorithms can easily be changed to reflect the potential of having different resolution in each subdomain, which makes it valuable as an adaptive algorithm. The same results are derived when the the alternate direction implicit (ADI) method is used as an efficient time-discretization scheme.

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.

Parallelizing implicit algorithms for time-dependent problems by parabolic domain decomposition

Applications of the multidomain Local Fourier Basis method [1], for the solution of PDEs on parallel computers are described. The present approach utilizes, in an explicit way, the rapid (exponential) decay of the fundamental solutions of elliptic operators resulting from semi-implicit discretizations of parabolic time-dependent problems. As a result, the global matching relations for the elemental solutions are decoupled into local interactions between pairs of solutions in neighboring domains. Such interactions require only local communications between processors with short communication links. Thus the present algorithm overcomes the global coupling, inherent both in the use of the spectral Fourier method and implicit time discretization scheme.

Highly Scalable Two- and Three-Dimensional Navier-Stokes Parallel Solvers on MIMD Multiprocessors

In this paper we present a new parallel algorithm for the solution of the incompressible two- and three-dimensional Navier-Stokes equations. The parallelization is achieved via domain decomposition. The computational region is considered in the form of a 2-D or 3-D periodic box decomposed into parallel strips (slabs). For time discretization we use a third order multistep method of [11]. The time discretization procedure results in solving global elliptic problems of (monotonic) Helmholtz and Poisson types in each time step. For the space discretization we employ the multidomain local Fourier (MDLF) method that was developed in [9, 10, 13]. The discretization in the periodic directions is performed by the standard Fourier method. In the direction across the strips we use the Local Fourier Basis technique which involves the overlapping of the neighboring subdomains and smoothing of local functions across the interior boundaries (interfaces). The matching of the local solutions is performed by adding properly weighted interface Greens functions. Their amplitudes are found in terms of the jumps of the solution and its first derivatives at the interfaces.The present paper extends the results of our previous works [1, 9, 10, 13] on parallel use of the MDLF method in three-fold aspects:1. In [1] a model Navier-Stokes type system was considered which does not include the pressure term. Correspondingly, in each time step only the Helmholtx type equations were solved. It was shown that the parallel solution of this equation can be accomplished using only local (neighbor-to-neighbor) communication due to localization properties of the Helmholtz operator. We consider the complete Navier-Stokes system including the pressure term. The solution of the Poisson equation for pressure has the potential to degrade the performance and the achieved speedup of a parallel algorithm due to the global nature of this equation that necessitates global communication among the processors. However, we show that only a few lowest harmonics require for the global data transfer whereas the rest of harmonics can be treated locally. Therefore, most of the communication that is required for parallelization of the Navier-Stokes solver using the MDLF method is mainly local between adjacent subdomains (processors). Moreover, the percentage of the time spent in global communication reduces as the size of the problem increases. Thus, the present parallel algorithm is highly scalable.2. In [l] we considered only 2-D equations. In this paper we extend the previous technique to 3-D problems.3. Previously, the MDLF solver was implemented only on the MEIKO parallel machine. In this paper the 2-D and 3-D Navier-Stokes solvers are implemented on three MIMD message-passing multiprocessors (a 60-processors IBM SP2, a 20-processors MOSIX [3], and a network of 10 Alpha workstations) and achieve an efficiency of more than 70% to 95%. The same code written with the PVM (parallel virtual machine [7]) software package was executed on all the above distinct computational platforms. Detailed performance results, which include scalability analysis, are presented.