Erik Schnetter

Alexander B. Atanasov, Erik Schnetter · harvard

We examine and extend Sparse Grids as a discretization method for partial differential equations (PDEs). Solving a PDE in $D$ dimensions has a cost that grows as $O(N^D)$ with commonly used methods. Even for moderate $D$ (e.g. $D=3$), this quickly becomes prohibitively expensive for increasing problem size $N$. This effect is known as the Curse of Dimensionality. Sparse Grids offer an alternative discretization method with a much smaller cost of $O(N \log^{D-1}N)$. In this paper, we introduce the reader to Sparse Grids, and extend the method via a Discontinuous Galerkin approach. We then solve the scalar wave equation in up to $6+1$ dimensions, comparing cost and accuracy between full and sparse grids. Sparse Grids perform far superior, even in three dimensions. Our code is freely available as open source, and we encourage the reader to reproduce the results we show.

Stamatis Vretinaris, Erik Schnetter

Formulating stable numerical methods for hyperbolic systems in curvilinear coordinate with singularities, e.g. spherical coordinates, is complicated by the presence of these singularities. We present a method for constructing high-order accurate, energy-stable finite difference operators satisfying the Summation-by-Parts (SBP) property on spherical domains, extending ideas presented by [C. Gundlach, J. M. Martín-García, and D. Garfinkle, CQG 30, 145003 (2013)]. We define discrete gradient and divergence operators that mirror the continuous integration-by-parts principle, even though there is a $1/r^p$ coordinate singularity present at the origin. We explicitly construct such operators up to order six. Our operators place a grid point directly on the origin. We also review how to construct stable SBP operators that straddle the origin. We analyze the accuracy and spectral radii of these operators, and we show example evolutions of the scalar wave equation to demonstrate the advantages of such operators.

Frank Löffler, Steven R. Brandt, Gabrielle Allen et al.

The Cactus Framework is an open-source, modular, portable programming environment for the collaborative development and deployment of scientific applications using high-performance computing. Its roots reach back to 1996 at the National Center for Supercomputer Applications and the Albert Einstein Institute in Germany, where its development jumpstarted. Since then, the Cactus framework has witnessed major changes in hardware infrastructure as well as its own community. This paper describes its endurance through these past changes and, drawing upon lessons from its past, also discusses future

Joanna Piotrowska, Jonah M. Miller, Erik Schnetter

Spectral methods provide an elegant and efficient way of numerically solving differential equations of all kinds. For smooth problems, truncation error for spectral methods vanishes exponentially in the infinity norm and $L_2$-norm. However, for non-smooth problems, convergence is significantly worse---the $L_2$-norm of the error for a discontinuous problem will converge at a sub-linear rate and the infinity norm will not converge at all. We explore and improve upon a post-processing technique---optimally convergent mollifiers---to recover exponential convergence from a poorly-converging spectral reconstruction of non-smooth data. This is an important first step towards using these techniques for simulations of realistic systems.