How We solve PDEs

Abstract

Time to solution I n t e n s i t y R a t e O p e r a t i o n s p e r b y t e t r a n s f e r r e d D Abstract. Important computational physics problems are often large-scale in nature, and it is highly desirable to have robust and high performing computational frameworks that can quickly address these problems. However, it is no trivial task to determine whether a computational framework is performing efficiently or is scalable. The aim of this paper is to present various strategies for better understanding the performance of any parallel computational frameworks for solving PDEs. Important performance issues that negatively impact time-to-solution are discussed, and we propose a performance spectrum analysis that can enhance one's understanding of critical aforementioned performance issues. As proof of concept, we examine commonly used finite element simulation packages and software and apply the performance spectrum to quickly analyze the performance and scalability across various hardware platforms, software implementations, and numerical dis-cretizations. It is shown that the proposed performance spectrum is a versatile performance model that is not only extendable to more complex PDEs such as hydrostatic ice sheet flow equations, but also useful for understanding hardware performance in a massively parallel computing environment. Potential applications and future extensions of this work are also discussed.

Scattered node compact finite difference-type formulas generated from radial basis functions

“…Consider the Lagrange form of s(x) (15). Note that we can expand φ( x − x i ) and L 2 φ( x − x i ) in powers of ε 2 so that the coefficients in front of these powers will be some polynomial in x.…”

“…However, this approach also raises the question of how the weights of the scattered node FD formulas should be computed. Abgrall [14] and Schönauer and Adolph [15], for example, propose extending the classical polynomial interpolation technique. This idea, however, has not become widely used, partly because it leads to several ambiguities about how to generate the FD formulas.…”

Adaptive meshless centres and RBF stencils for Poisson equation

“…Note that the best known approach to the generation of finite difference stencils on irregular centres is the polynomial least squares method [19,23,38,3]. This and related methods are often understood as local collocation.…”

“…In contrast to this, generalised finite difference methods allow a lot of flexibility in the choice of stencil supports. For the polynomial methods, sophisticated algorithms have been developed, see [38,20,35]. We are not aware, however, of any work in this direction for the RBF based generalised finite differences.…”

A singularity-avoiding moving least squares scheme for two-dimensional unstructured meshes

“…However, no further discussion is provided of the general conditions under which singularities occur. The singularity problem for moving least squares has also been noted by Netuzhylov [4] and Prax et al [5], and for a similar problem by Schoenauer and Adolph [6], who also observed the occurrence of singularities when the data points lie along straight lines. However, to the best knowledge of the authors, a general theory for predicting the occurrence of singularities in moving least squares has never been published.…”