arXiv subjects

Phillip Colella

Publications and source records attributed to Phillip Colella.

12 recordsLinked to original sources

arXiv2023

ProtoX: A First Look

We present a first look at ProtoX, a code generation framework for stencil and pointwise operations that occur frequently in the numerical solution of partial differential equations. ProtoX has Proto as its library frontend and SPIRAL as the backend. Proto is a C++ based domain specific library which optimizes the algorithms used to compute the numerical solution of partial differential equations. Meanwhile, SPIRAL is a code generation system that focuses on generating highly optimized target code. Although the current design layout of Proto and its high level of abstractions provide a user friendly set up, there is still a room for improving it's performance by applying various techniques either at a compiler level or at an algorithmic level. Hence, in this paper we propose adding SPIRAL as the library backend for Proto enabling abstraction fusion, which is usually difficult to perform by any compiler. We demonstrate the construction of ProtoX by considering the 2D Poisson equation as a model problem from Proto. We provide the final generated code for CPU, Multi-core CPU, and GPU as well as some performance numbers for CPU.

cs.MS↗

arXiv2018

An adaptive local discrete convolution method for the numerical solution of Maxwell's equations

We present a numerical method for solving the free-space Maxwell's equations in three dimensions using compact convolution kernels on a rectangular grid. We first rewrite Maxwell's Equations as a system of wave equations with auxiliary variables and discretize its solution from the method of spherical means. The algorithm has been extended to be used on a locally-refined nested hierarchy of rectangular grids.

math.NA↗

arXiv2017

High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry

We present a high-order spatial discretization of a continuum gyrokinetic Vlasov model in axisymmetric tokamak edge plasma geometries. Such models describe the phase space advection of plasma species distribution functions in the absence of collisions. The gyrokinetic model is posed in a four-dimensional phase space, upon which a grid is imposed when discretized. To mitigate the computational cost associated with high-dimensional grids, we employ a high-order discretization to reduce the grid size needed to achieve a given level of accuracy relative to lower-order methods. Strong anisotropy induced by the magnetic field motivates the use of mapped coordinate grids aligned with magnetic flux surfaces. The natural partitioning of the edge geometry by the separatrix between the closed and open field line regions leads to the consideration of multiple mapped blocks, in what is known as a mapped multiblock (MMB) approach. We describe the specialization of a more general formalism that we have developed for the construction of high-order, finite-volume discretizations on MMB grids, yielding the accurate evaluation of the gyrokinetic Vlasov operator, the metric factors resulting from the MMB coordinate mappings, and the interaction of blocks at adjacent boundaries. Our conservative formulation of the gyrokinetic Vlasov model incorporates the fact that the phase space velocity has zero divergence, which must be preserved discretely to avoid truncation error accumulation. We describe an approach for the discrete evaluation of the gyrokinetic phase space velocity that preserves the divergence-free property to machine precision.

cs.CE↗

arXiv2017

Computation of Volume Potentials on Structured Grids Using the Method of Local Corrections

We present a new version of the Method of Local Corrections (MLC) \cite{mlc}, a multilevel, low communications, non-iterative, domain decomposition algorithm for the numerical solution of the free space Poisson's equation in 3D on locally-structured grids. In this method, the field is computed as a linear superposition of local fields induced by charges on rectangular patches of size $O(1)$ mesh points, with the global coupling represented by a coarse grid solution using a right-hand side computed from the local solutions. In the present method, the local convolutions are further decomposed into a short-range contribution computed by convolution with the discrete Green's function for an $Q^{th}$-order accurate finite difference approximation to the Laplacian with the full right-hand side on the patch, combined with a longer-range component that is the field induced by the terms up to order $P-1$ of the Legendre expansion of the charge over the patch. This leads to a method with a solution error that has an asymptotic bound of $O(h^P) + O(h^Q) + O(\epsilon h^2) + O(\epsilon)$, where $h$ is the mesh spacing, and $\epsilon$ is the max norm of the charge times a rapidly-decaying function of the radius of the support of the local solutions scaled by $h$. Thus we have eliminated the low-order accuracy of the original method (which corresponds to $P=1$ in the present method) for smooth solutions, while keeping the computational cost per patch nearly the same with that of the original method. Specifically, in addition to the local solves of the original method we only have to compute and communicate the expansion coefficients of local expansions (that is, for instance, 20 scalars per patch for $P=4$). Several numerical examples are presented to illustrate the new method and demonstrate its convergence properties.

math.NA↗

arXiv2016

A Survey of High Level Frameworks in Block-Structured Adaptive Mesh Refinement Packages

Over the last decade block-structured adaptive mesh refinement (SAMR) has found increasing use in large, publicly available codes and frameworks. SAMR frameworks have evolved along different paths. Some have stayed focused on specific domain areas, others have pursued a more general functionality, providing the building blocks for a larger variety of applications. In this survey paper we examine a representative set of SAMR packages and SAMR-based codes that have been in existence for half a decade or more, have a reasonably sized and active user base outside of their home institutions, and are publicly available. The set consists of a mix of SAMR packages and application codes that cover a broad range of scientific domains. We look at their high-level frameworks, and their approach to dealing with the advent of radical changes in hardware architecture. The codes included in this survey are BoxLib, Cactus, Chombo, Enzo, FLASH, and Uintah.

cs.DC↗

arXiv2016

A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov-Poisson Equation

Numerical solutions to the Vlasov-Poisson system of equations have important applications to both plasma physics and cosmology. In this paper, we present a new Particle-in-Cell (PIC) method for solving this system that is 4th-order accurate in both space and time. Our method is a high-order extension of one presented previously [B. Wang, G. Miller, and P. Colella, SIAM J. Sci. Comput., 33 (2011), pp. 3509--3537]. It treats all of the stages of the standard PIC update - charge deposition, force interpolation, the field solve, and the particle push - with 4th-order accuracy, and includes a 6th-order accurate phase-space remapping step for controlling particle noise. We demonstrate the convergence of our method on a series of one- and two- dimensional electrostatic plasma test problems, comparing its accuracy to that of a 2nd-order method. As expected, the 4th-order method can achieve comparable accuracy to the 2nd-order method with many fewer resolution elements.

math.NA↗

arXiv2015

The Convergence of Particle-in-Cell Schemes for Cosmological Dark Matter Simulations

Particle methods are a ubiquitous tool for solving the Vlasov-Poisson equation in comoving coordinates, which is used to model the gravitational evolution of dark matter in an expanding universe. However, these methods are known to produce poor results on idealized test problems, particularly at late times, after the particle trajectories have crossed. To investigate this, we have performed a series of one- and two-dimensional "Zel'dovich Pancake" calculations using the popular Particle-in-Cell (PIC) method. We find that PIC can indeed converge on these problems provided the following modifications are made. The first modification is to regularize the singular initial distribution function by introducing a small but finite artificial velocity dispersion. This process is analogous to artificial viscosity in compressible gas dynamics, and, as with artificial viscosity, the amount of regularization can be tailored so that its effect outside of a well-defined region - in this case, the high-density caustics - is small. The second modification is the introduction of a particle remapping procedure that periodically re-expresses the dark matter distribution function using a new set of particles. We describe a remapping algorithm that is third-order accurate and adaptive in phase space. This procedure prevents the accumulation of numerical errors in integrating the particle trajectories from growing large enough to significantly degrade the solution. Once both of these changes are made, PIC converges at second order on the Zel'dovich Pancake problem, even at late times, after many caustics have formed. Furthermore, the resulting scheme does not suffer from the unphysical, small-scale "clumping" phenomenon known to occur on the Pancake problem when the perturbation wave vector is not aligned with one of the Cartesian coordinate axes.

astro-ph.CO↗

arXiv2015

A Single Stage Flux-Corrected Transport Algorithm for High-Order Finite-Volume Methods

We present a new limiter method for solving the advection equation using a high-order, finite-volume discretization. The limiter is based on the flux-corrected transport algorithm. We modify the classical algorithm by introducing a new computation for solution bounds at smooth extrema, as well as improving the pre-constraint on the high-order fluxes. We compute the high-order fluxes via a method of lines approach with fourth order Runge-Kutta as the time integrator. For computing low-order fluxes, we select the corner transport upwind method due to its improved stability over donor-cell upwind. Several spatial differencing schemes are investigated for the high-order flux computation, including centered difference and upwind schemes. We show that the upwind schemes perform well on account of the dissipation of high wavenumber components. The new limiter method retains high-order accuracy for smooth solutions and accurately captures fronts in discontinuous solutions. Further, we need only apply the limiter once per complete time step.

math.NA↗

arXiv2011

Numerical Implementation of Streaming Down the Gradient: Application to Fluid Modeling of Cosmic Rays and Saturated Conduction

The equation governing the streaming of a quantity down its gradient superficially looks similar to the simple constant velocity advection equation. In fact, it is the same as an advection equation if there are no local extrema in the computational domain or at the boundary. However, in general when there are local extrema in the computational domain it is a non-trivial nonlinear equation. The standard upwind time evolution with a CFL-limited time step results in spurious oscillations at the grid scale. These oscillations, which originate at the extrema, propagate throughout the computational domain and are undamped even at late times. These oscillations arise because of unphysically large fluxes leaving (entering) the maxima (minima) with the standard CFL-limited explicit methods. Regularization of the equation shows that it is diffusive at the extrema; because of this, an explicit method for the regularized equation with $Δt \propto Δx^2$ behaves fine. We show that the implicit methods show stable and converging results with $Δt \propto Δx$; however, surprisingly, even implicit methods are not stable with large enough timesteps. In addition to these subtleties in the numerical implementation, the solutions to the streaming equation are quite novel: non-differentiable solutions emerge from initially smooth profiles; the solutions show transport over large length scales, e.g., in form of tails. The fluid model for cosmic rays interacting with a thermal plasma (valid at space scales much larger than the cosmic ray Larmor radius) is similar to the equation for streaming of a quantity down its gradient, so our method will find applications in fluid modeling of cosmic rays.

astro-ph.HE↗

arXiv2009

Extremum-Preserving Limiters for MUSCL and PPM

Limiters are nonlinear hybridization techniques that are used to preserve positivity and monotonicity when numerically solving hyperbolic conservation laws. Unfortunately, the original methods suffer from the truncation-error being first-order accurate at all extrema despite the accuracy of the higher-order method. To remedy this problem, higher-order extensions were proposed that relied on elaborate analytic and geometric constructions. Since extremum-preserving limiters are applied only at extrema, additional computational cost is negligible. Therefore, extremum-preserving limiters ensure higher-order spatial accuracy while maintaining simplicity. This report presents higher-order limiting for (i) computing van Leer slopes and (ii) adjusting parabolic profiles. This limiting preserves monotonicity and accuracy at smooth extrema, maintains stability in the presence of discontinuities and under-resolved gradients, and is based on constraining the interpolated values at extrema (and only at extrema) by using nonlinear combinations of second derivatives. The van Leer limiting can be done separately and implemented in MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) or done in concert with the parabolic profile limiting and implemented in PPM (Piecewise Parabolic Method). The extremum-preserving limiters elegantly fit into any algorithm which uses conventional limiting techniques. Limiters are outlined for scalar advection and nonlinear systems of conservation laws. This report also discusses the fourth-order correction to the point-valued, cell-centered initial conditions that is necessary for implementing higher-order limiting.

physics.comp-ph↗

arXiv2007

Block Structured Adaptive Mesh and Time Refinement for Hybrid, Hyperbolic + N-body Systems

We present a new numerical algorithm for the solution of coupled collisional and collisionless systems, based on the block structured adaptive mesh and time refinement strategy (AMR). We describe the issues associated with the discretization of the system equations and the synchronization of the numerical solution on the hierarchy of grid levels. We implement a code based on a higher order, conservative and directionally unsplit Godunov's method for hydrodynamics; a symmetric, time centered modified symplectic scheme for collisionless component; and a multilevel, multigrid relaxation algorithm for the elliptic equation coupling the two components. Numerical results that illustrate the accuracy of the code and the relative merit of various implemented schemes are also presented.

astro-ph↗

arXiv2006

A Modified Higher Order Godunov's Scheme for Stiff Source Conservative Hydrodynamics

Hyperbolic conservation laws with stiff source terms appear in the study of a variety of physical systems. Early work showed that the use of formally second-order accurate semi-implicit methods could lead to a substantial loss of accuracy, due to inconsistencies between the flux calculation without sources and the limiting equilibrium behavior of the gas. In this paper we present an efficient second order accurate scheme to treat stiff source terms within the framework of higher order Godunov's methods. We employ Duhamel's formula to devise a modified predictor step which accounts for the effects of stiff source terms on the conservative fluxes and recovers the correct isothermal behavior in the limit of an infinite cooling/reaction rate. Source term effects on the conservative quantities are fully accounted for by means of a one-step, second order accurate semi-implicit corrector scheme based on the deferred correction method of Dutt et. al. We demostrate the accurate, stable and convergent results of the proposed method through a set of benchmark problems for a variety of stiffness conditions and source types.

astro-ph↗