Software For Partial Differential Equations Research Articles

Performance analysis of error-control B-spline Gaussian collocation software for PDEs

B-spline Gaussian collocation software has been widely used in the numerical solution of boundary value ordinary differential equations (BVODEs) and 1D partial differential equations (PDEs) for several decades. Such packages represent the numerical solution in terms of a piecewise polynomial (B-spline) basis with basis coefficients determined through the use of Gaussian collocation. The software package, BACOL, developed over a decade ago, was the first 1D PDE package of this type to provide both temporal and spatial error control. A recently developed package, BACOLI, improves upon the efficiency of BACOL through the use of new types of spatial error estimation and control. The complexity of the interactions among the component numerical algorithms used by these packages (particularly the spatial and temporal error estimation and control algorithms) implies that extensive testing and analysis of the test results is an essential factor in the ongoing development of these packages In this paper, we investigate the performance of BACOL and BACOLI with respect to several important machine independent algorithmic measures, examine the effectiveness of the new spatial error estimation and control strategies, and investigate the influence of the choice of the degree of the B-spline basis on the performance of the solvers. These results will provide new insights into how to improve BACOLI, potentially lead to improvements in Gaussian collocation BVODE solvers, and guide further development of B-spline Gaussian collocation software with error control for 2D PDEs.

Modeling turbulent flow from dam break using slow manifolds

We present a novel approach based on centre manifold techniques to describe the large scale dynamics of the mean turbulent dam-break flow. We avoid empirical assumptions about the cross-stream profile of the velocity; instead a solution is obtained using free surface and bed boundary conditions that accommodate constant turbulent shear as an approximately neutral mode. We describe the turbulent dynamics across the flow, and identify important factors affecting the turbulent dissipation in the lateral direction. Available experimental data verify the results. References J. Carr. Applications of centre manifold theory, volume 35 of Applied Mathematical Sciences. Springer--Verlag, 1981. D. J. Georgiev, A. J. Roberts, and D. V. Strunin. The dynamics of the vertical structure of turbulence in flood flows. ANZIAM Journal, 48(CTAC-2006):C573--C590, 2007. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/124. D. J. Georgiev, A. J. Roberts, and D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: $k$-$\omega $ model. Discrete and Continuous Dynamical Systems Supplements, (Special):419--428, 2007. http://aimsciences.org/journals/pdfs.jsp?paperID=2848&mode=full. A. J. Hogg and D. Pritchard. The effects of hydraulic resistance on dam-break and other shallow inertial flows. Journal of Fluid Mechanics, 501:179--212, 2004. I. M. Janosi, D. Jan, K. G. Szabo, and T. Tel. Turbulent drag reduction in dam-break flows. Experiments in Fluids, 37:219--229, 2004. P. Keast and P. H. Muir. {EPDCOL}: A more efficient {PDECOL} code. ACM Transactions on Mathematical Software, 17(2):153--166, 1991. Y. A. Kuznetsov. Elements of applied bifurcation theory, volume 112 of Applied Mathematical Sciences. Springer--Verlag, 1995. N. K. Madsen and R. F. Sincovec. {PDECOL}: General collocation software for partial differential equations. ACM Transactions on Mathematical Software, 5(3):326--351, 1979. I. Nezu. Open-channel flow turbulence and its research prospects in the 21st century. Journal of Hydraulic Engineering, 131(4):229--246, April 2005. T. M. Oezgoekmen, T. Iliescu, P. F. Fischer, A. Srinivasan, and J. Duan. Large eddy simulation of stratified mixing and two-dimensional dam-break problem in a rectagular enclosed domain. Ocean Modelling, 16:106--140, 2007. A. J. Roberts. Low-dimensional modelling of dynamics via computer algebra. Computer Physics Communications, 100:215--230, 1997. A. J. Roberts. Computer algebra describes flow of turbulent floods via the Smagorinsky large eddy closure. Technical report, University of Southern Queensland, 2008. http://eprints.usq.edu.au/4008/. A. J. Roberts, D. J. Georgiev, and D. V. Strunin. Model turbulent floods with the Smagorinski large eddy closure. http://arxiv.org/abs/0805.3192/. P. K. Stansby, A. Chegini, and T. C. D. Barnes. The initial stages of dam-break flow. Journal of Fluid Mechanics, 374:407--424, 1998.

Performance of embedded boundary methods for CFD with complex geometry

In this paper, we discuss some of the issues in obtaining high performance for block-structured adaptive mesh refinement software for partial differential equations in complex geometry using embedded boundary/volume-of-fluid methods. We present the design of an adaptive embedded boundary multigrid algorithm for elliptic problems. We show examples in which this new elliptic solver scales to 1000 processors. We also apply this technology to more complex mathematical and physical algorithms for incompressible fluid dynamics and demonstrate similar scaling.

Embedded boundary algorithms and software for partial differential equations

In this paper, we give an overview of a set of methods being developed for solving classical PDEs in irregular geometries, or in the presence of free boundaries. In this approach, the irregular geometry is represented on a rectangular grid by specifying the intersection of each grid cell with the region on one or the other side of the boundary. This leads to a natural conservative discretization of the solution to the PDE on either side of the boundary. Stable and robust hyperbolic and linear elliptic/parabolic solvers have been designed and implemented. Example applications of this approach are shown for compressible and incompressible gas dynamics problems in complex geometries, and for surface diffusion in a cell membrane.

Performance and scaling of locally-structured grid methods for partial differential equations

In this paper, we discuss some of the issues in obtaining high performance for block-structured adaptive mesh refinement software for partial differential equations. We show examples in which AMR scales to thousands of processors. We also discuss a number of metrics for performance and scalability that can provide a basis for understanding the advantages and disadvantages of this approach.

A Survey of Software for Partial Differential Equations

A Survey of Software for Partial Differential Equations

Factors limiting academic participation in the development of large mathematical software packages

The December 1975 SIGNUM meeting of Software for Partial Differential Equations seemed to formalize a trend which over the past few years has become ever more evident. Namely, the the development of medium and large scale software packages, say anything more than five hundred statements, is a project which can no longer be carried out in a university. Such projects are now almost wholly within the province of industrial and governmental laboratories. As can be seen from the program of the meeting (ACM SIGNUM Newsletter 10, 1975; No. 4, pp. 2-3), only two our of seventeen papers came from universities and one of these two papers concerned itself with a modest size algorithm for solving systems of linear algebraic equations. The reasons for this state of affairs are not hard to find. It is expensive to develop such software, and it is not altogether appropriate in a university to tie up manpower and money in such extensive undertakings. After all, departments of mechanical engineering do not build boilers, or perhaps, since new program development involves much innovation, a more accurate analogy would be to say that departments of aeronautical engineering do not build new jetliners. Also, because of tight money and excess personnel, pressures on faculty to "Produce" have increased and producing means publishing.