Phillip Colella

• Lawrence Berkeley National Laboratory

We present the HelioCubed, a high-order magnetohydrodynamic (MHD) code designed for modeling the inner heliosphere. The code is designed to achieve 4th order accuracy both in space and in time. In addition, HelioCubed can perform simulations on mapped grids, such as those based on cubed spheres, which makes it possible to overcome stability limitat...

To address Objective II of the National Space Weather Strategy and Action Plan “Develop and Disseminate Accurate and Timely Space Weather Characterization and Forecasts” and US Congress PROSWIFT Act 116–181, our team is developing a new set of open-source software that would ensure substantial improvements of Space Weather (SWx) predictions. On the...

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 num...

View Video Presentation: https://doi.org/10.2514/6.2022-2322.vid In this paper, we show an implementation of the signed distance function for tracking an interface between two fluids with an arbitrary geometry. The objective of this study is to create an algorithm to implicitly track the boundary of an additive manufacturing material in a direct in...

We present a new version of the method of local corrections (MLC) of Mc- Corquodale, Colella, Balls, and Baden (2007), a multilevel, low-communication, noniterative domain decomposition algorithm for the numerical solution of the free space Poisson's equation in three dimensions on locally structured grids. In this method, the field is computed as...

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 us...

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 wh...

GPUs, with their high bandwidths and computational capabilities are an increasingly popular target for scientific computing. Unfortunately, to date, harnessing the power of the GPU has required use of a GPU-specific programming model like CUDA, OpenCL, or OpenACC. As such, in order to deliver portability across CPU-based and GPU-accelerated superco...

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 ch...

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...

Adaptive mesh refinement (AMR) is a technique that has been featured only sporadically in atmospheric science literature. This paper aims to demonstrate the utility of AMR for simulating atmospheric flows. Several test cases are implemented in a 2D shallow-water model on the sphere using the Chombo-AMR dynamical core. This high-order finite-volume...

A new method for solving the transverse part of the free-space Maxwell equations in three dimensions is presented. By taking the Helmholtz decomposition of the electric field and current sources and considering only the divergence-free parts, we obtain an explicit real-space representation for the transverse propagator that explicitly respects fini...

We present an approach to designing arbitrarily high-order finite volume spatial discretizations on locally-rectangular grids. It is based on the use of a simple class of high-order quadratures for computing the average of fluxes over faces. This approach has the advantage of being a variation on widely-used second-order methods, so that the prior...

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,...

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,...

A fourth-order accurate finite-volume method is presented for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space are combined with detailed mechanisms for accommodating the adap...

We present a high-order finite-volume approach for solving the shallow-water equations on the sphere, using multiblock grids on the cubed sphere. This approach combines a Runge-Kutta time discretization with a fourth-order-accurate spatial discretization and includes adaptive mesh refinement and refinement in time. Results of tests show fourth-orde...

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 fluxe...

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 fluxe...

We present an approach to solving hyperbolic conservation laws by finite-volume methods on mapped multiblock grids, extending the approach of Colella, Dorr, Hittinger, and Martin (2011) [10] for grids with a single mapping. We consider mapped multiblock domains for mappings that are conforming at inter-block boundaries. By using a smooth continuati...

We present an algorithm to produce the necessary geometric information for finite volume calculations in the context of Cartesian grids with embedded boundaries. Given an order of accuracy for the overall calculation, we show what accuracy is required for each of the geometric quantities and we demonstrate how to calculate the moments using the div...

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 cros...

A lattice-Boltzmann model to solve the equivalent of the Navier–Stokes equations on adaptively refined grids is presented. A method for transferring information across interfaces between different grid resolutions was developed following established techniques for finite-volume representations. This new approach relies on a space–time interpolation...

This study focuses on a fourth-order boundary treatment for finite-volume schemes to solve the compressible Navier-Stokes equations on a Cartesian grid. A fourth-order finite-volume stencil is derived for the viscous stress tensor operator and the modified fourth-order stencil near the physical boundary is developed. Fourier error analysis and stab...

We present an unsplit method for the time-dependent compressible Navier-Stokes equations in two and three dimensions. We use a conservative, second-order Godunov algorithm. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We solve for viscous and conductive terms with a second-order semiimplicit algorithm. We demonst...

This paper describes the numerical simulation of a shock wave refracting at a gas interface. Shock tube experiments performed by Abd-el-Fattah and Henderson using a multifluid, adaptive mesh refinement algorithm have been duplicated. The results of four of these calculations and comparison to the shock tube experiments is reported. The goal of this...

Geodesic acoustic modes (GAMs) are an important phenomenon in a tokamak edge plasma. They regulate turbulence in a low confinement (L-mode) regime and can play an important role in the low to high (L–H) mode transition. It is therefore of considerable importance to develop a detailed theoretical understanding of their dynamics and relaxation proces...

The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v∥, μ) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v∥ and μ are...

A conservative lattice-Boltzmann method is presented for solving the time-dependent Navier-Stokes equations at low Mach numbers on lattices that are adaptively refined in space and time. A method for coupling the interfaces between grids at different resolutions was constructed following techniques established for finite-volume computational fluid...

The numerical solution of high dimensional Vlasov equation is usually performed by particle-in-cell (PIC) methods. However, due to the well-known numerical noise, it is challenging to use PIC methods to get a precise description of the distribution function in phase space. To control the numerical error, we introduce an adaptive phase-space remappi...

COGENT is a continuum gyrokinetic code being developed by the Edge Simulation Laboratory for edge plasmas. The code is distinguished by the use of a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of divertor geometry without...

COGENT is a continuum gyrokinetic code for edge plasmas being developed by the Edge Simulation Laboratory collaboration. The code is distinguished by application of the fourth order conservative discretization, and mapped multiblock grid technology to handle the geometric complexity of the tokamak edge. It is written in v‖-μ (parallel velocity – ma...

We present a fourth-order accurate algorithm for solving Poisson's equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mes...

COGENT is a full-f continuum kinetic code being developed for study of edge physics phenomena in tokamaks. The code is distinguished by 4th order conservative discretization and mapped multiblock grid technology to handle the geometric complexity of the tokamak edge. We discuss a number of recent neoclassical results in closed-flux-surface geometry...

A fourth-order accurate finite-volume method is presented for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space combined with detailed mechanisms for accommodating the adapting...

We present a description of the adaptive mesh refinement (AMR) implementation of the PLUTO code for solving the equations of classical and special relativistic magnetohydrodynamics (MHD and RMHD). The current release exploits, in addition to the static grid version of the code, the distributed infrastructure of the CHOMBO library for multidimension...

Adaptive mesh refinement (AMR) applications to solve partial differential equations (PDE) are very challenging to scale efficiently to the petascale regime. We describe optimizations to the Chombo AMR framework that enable it to scale efficiently to petascale on the Cray XT5. We describe an example of a hyperbolic solver (inviscid gas dynamics) an...

We describe a two-dimensional shallow water model designed to simulate water quality and flooding. The model uses a finite-volume discretization of the shallow water equations on an adaptive Cartesian mesh, using embedded boundaries to represent complex topography. For flooding applications, we use adaptive mesh refinement (AMR) to evolve Cartesian...

We present an approach for constructing finite-volume methods of any order of ac-curacy for control-volume discretizations of space defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the fl...

We present a method for solving Poisson and heat equations with discontinuous coefficients in two- and three-dimensions. It uses a Cartesian cut-cell/embedded boundary method to represent the interface between materials, as described in Johansen and Colella (1998). Matching conditions across the interface are enforced using an approximation to flux...

We present a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement. The underlying method is a generalization of that in [5] to nonlinear systems, and is based on using fourth-order accurate quadratures for computing fluxes on faces, combin...

We present a new accurate and efficient particle-in-cell (PIC) method for computing the dynamics of one-dimensional kinetic plasmas. The method overcomes the numerical noise inherent in particle-based methods by periodically remapping the distribution function on a hierarchy of locally refined grids in phase space. Remapping on phase-space grids al...

Adaptive mesh refinement (AMR) is an efficient technique for solving systems of partial differential equations numerically. The underlying algorithm determines where and when a base spatial and temporal grid must be resolved further in order to achieve the desired precision and accuracy in the numerical solution. However, propagating wave solutions...

COGENT is a continuum gyrokinetic code being developed by the Edge Simulation Laboratory for edge plasmas. The code is distinguished by application of 4th order conservative discretization, and mapped multiblock grid technology to handle the geometric complexity of the tokamak edge. We report on a verification campaign involving simulation of geode...

We present an all-speed algorithm for magneto-hydrodynamics (MHD) similar to the work of Colella & Pao (J. Comput. Phys. 1999) for low speed hydrodynamics. The method is based on an asymptotic ordering of scales relevant for tokamak MHD physics. The central idea is to Hodge decompose the velocity, and a splitting of the magnetic guide field analogo...

Capillary plasma channels are used to extend the propagation distance of relativistically intense laser pulses for laser plasma acceleration [1], and axial density modulation has been used to stabilize injection at LBNL. Channel formation is a complex process in which a gas is ionized via a slow discharge, and subsequently stabilized by a capillary...

We describe a two-dimensional shallow water model whose initial implementation simulates flows in the San Francisco Bay and Sacramento-San Joaquin Delta. This model, called REALM, is based on a Cartesian grid, embedded boundary discretization of the shallow water equations. We employ parallel computation and adaptive mesh refinement for rapid compu...

We present a numerical method for computing the signed distance to a piecewise-smooth surface dened as the zero set of a function. It is based on a marching method by Kim (Kim01) and a hybrid discretization of rst- and second-order discretizations of the signed distance function equation. If the solution is smooth at a point and at all of the point...

We analyze the source of the self-force errors in the node-centered adaptive-mesh-refinement particle-in-cell (AMR-PIC) algorithm and propose a method for reducing those self-forces. Our approach is based on a method of charge deposition due to Mayo [A. Mayo, The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM Jou...

This paper presents a conservative front-tracking method for shocks and contact discontinuities that is second-order accurate. It is based on a volume-of-fluid method that treats the moving front with concepts similar to those of an embedded-boundary method. Special care is taken in the centering of the data to ensure the right order of accuracy at...

We describe our progress in the development of a fourth-order, finite-volume discretization of a nonlinear, full-f gyrokinetic Vlasov-Poisson system in mapped coordinates. The approach treats the configuration and velocity components of phase space on an equal footing, using a semi-discretization with limited centered fluxes combined with a fourth-...

Gravity waves arise in gravitationally stratified compressible flows at low Mach and Froude numbers, and these waves impose a sharp restriction on the time step. This paper presents a filtering strategy for the fully compressible equations based on normal-mode analysis that is used throughout the simulation to compute the fast dynamics and is able...

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 doma...

We are developing a new class of finite-volume methods on locally-refined and mapped grids, which are at least fourth-order accurate in regions where the solution is smooth. This paper discusses the implementation of such methods for time-dependent problems on both Cartesian and mapped grids with adaptive mesh refinement. We show 2D results with th...

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable-density, and within arbitrarily complex and possibly anisotropic domains. The method is new because i...

We present a conservative finite difference method designed to capture elastic wave propagation in viscoelastic fluids in two dimensions. We model the incompressible Navier-Stokes equations with an extra viscoelastic stress described by the Oldroyd-B constitutive equations. The equations are cast into a hybrid conservation form which is amenable to...

We present edge gyrokinetic simulations of tokamak plasmas using the fully non-linear (full-f) continuum code TEMPEST. A non-linear Boltzmann model is used for the electrons. The electric field is obtained by solving the 2D gyrokinetic Poisson equation. We demonstrate the following. (1) High harmonic resonances (n > 2) significantly enhance geodesi...

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-or...

We describe recent developments in high-order (greater than second-order) accurate finite-volume discretizations of partial differential equations (PDE) in divergence form. We will address a number of algorithmic issues that arise in these methods, including the choice of limiters for hyperbolic problems that preserve high-order accuracy; defining...

In this paper, we discuss some of the issues in obtaining high performance for block-structured adaptive mesh refinement software for Poisson's equation. 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 d...

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 doma...

The Edge Simulation Laboratory (ESL) is a multi-institutional collaboration to develop kinetic edge codes using continuum techniques. A new code, based on fourth-order conservative finite-volume discretization of gyrokinetic equations, has recently become operational. Initially the code is electrostatic, 4D (axisymmetric), with a Miller (core, shap...

We present edge gyrokinetic neoclassical simulations of tokamak plasmas using the fully nonlinear (full-f) continuum code TEMPEST. A nonlinear Boltzmann model is used for the electrons. The electric field is obtained by solving the 2D gyrokinetic Poisson Equation. We demonstrate the following: (1) High harmonic resonances (n > 2) significantly enha...

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. Thi...

To construct finite-volume methods for PDEs in arbitrary dimension to arbitrary accuracy in the presence of irregular boundaries, we show that estimates of moments, integrals of monomials, over various regions are all that are needed. If implicit functions are used to represent the irregular boundary, the needed moments can be computed straightforw...

We present a new limiter for the PPM method of Colella and Woodward [P. Colella, P.R. Woodward, The Piecewise Parabolic Method (PPM) for gas-dynamical simulations, Journal of Computational Physics 54 (1984) 174–201] that preserves accuracy at smooth extrema. It is based on constraining the interpolated values at extrema (and only at extrema) using...

We regularize the variable coefficient Helmholtz equations arising from implicit time discretizations for resistive MHD, in a way that leads to a symmetric positive-definite system uniformly in the time step. Standard centered-difference discret- izations in space of the resulting PDE leads to a method that is second-order accurate, and that can be...