High-Order Finite-Volume Schemes for Magnetohydrodynamics

New high-order finite-volume numerical schemes for the magnetohydrodynamics equations are proposed in two and three dimensions. Two different sets of magnetohydrodynamics equations are considered. The first set is the ideal magnetohydrodynamics system, which assumes that the fluid can be treated as a perfect conductor. The second set is resistive MHD, which involves non-zero resistivity. A high-order central essentially nonoscillatory (CENO) approach is employed, which combines unlimited k-exact polynomial reconstruction with a monotonicity preserving scheme. The CENO schemes, which were originally developed for compressible fluid flow, are applied to the MHD equations, along with two possible control mechanisms for divergence error of the magnetic field. The hyperbolic fluxes are calculated by solving a Riemann problem at each cell interface, and elliptic fluxes are computed through k-exact gradient interpolation where point-wise values of the gradients are required. Smooth test problems and test cases with discontinuities (weak or strong) are considered, and convergence studies are presented for both the ideal and resistive MHD systems. Several potential space physics applications are explored. For these simulations, cubed-sphere grids are used to model the interaction of the solar wind with planetary bodies or their satellites. The basic cubed-sphere grid discretizes a simulation domain between two concentric spheres using six root blocks (corresponding to the six faces of a cube). Conditions describing the atmosphere of the inner body can be applied at the boundary of the inner sphere. For some problems we also need to solve equations within the inner sphere, for which we develop a seven-block cubed-sphere grid where the empty space inside the interior sphere is discretized as a seventh root block. We consider lunar flow problems for which we employ the seven-block cubed-sphere mesh. Ideal MHD is solved between the inner and outer spheres of the grid, and the magnetic diffusion equations are solved within the inner sphere, which represents the lunar interior. Two cases are considered: one is without intrinsic magnetic field, where only a wake is expected without any bow shock forming ahead of the Moon, and the second is with a small dipole moment to model a lunar crustal magnetic anomaly, in which case a small-scale magnetosphere is expected ahead of the region with the magnetic anomaly

Multi-patch methods in general relativistic astrophysics - I. Hydrodynamical flows on fixed backgrounds

Many systems of interest in general relativistic astrophysics, including neutron stars, accreting compact objects in X-ray binaries and active galactic nuclei, core collapse, and collapsars, are assumed to be approximately spherically symmetric or axisymmetric. In Newtonian or fixed-background relativistic approximations it is common practice to use spherical polar coordinates for computational grids; however, these coordinates have singularities and are difficult to use in fully relativistic models. We present, in this series of papers, a numerical technique which is able to use effectively spherical grids by employing multiple patches. We provide detailed instructions on how to implement such a scheme, and present a number of code tests for the fixed background case, including an accretion torus around a black hole.Comment: 26 pages, 20 figures. A high-resolution version is available at http://www.cct.lsu.edu/~bzink/papers/multipatch_1.pd

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

Three-dimensional general-relativistic hydrodynamic simulations of binary neutron star coalescence and stellar collapse with multipatch grids

We present a new three-dimensional, general-relativistic hydrodynamic evolution scheme coupled to dynamical spacetime evolutions which is capable of efficiently simulating stellar collapse, isolated neutron stars, black hole formation, and binary neutron star coalescence. We make use of a set of adapted curvilinear grids (multipatches) coupled with flux-conservative, cell-centered adaptive mesh refinement. This allows us to significantly enlarge our computational domains while still maintaining high resolution in the gravitational wave extraction zone, the exterior layers of a star, or the region of mass ejection in merging neutron stars. The fluid is evolved with a high-resolution, shock-capturing finite volume scheme, while the spacetime geometry is evolved using fourth-order finite differences. We employ a multirate Runge-Kutta time-integration scheme for efficiency, evolving the fluid with second-order integration and the spacetime geometry with fourth-order integration. We validate our code by a number of benchmark problems: a rotating stellar collapse model, an excited neutron star, neutron star collapse to a black hole, and binary neutron star coalescence. The test problems, especially the latter, greatly benefit from higher resolution in the gravitational wave extraction zone, causally disconnected outer boundaries, and application of Cauchy-characteristic gravitational wave extraction. We show that we are able to extract convergent gravitational wave modes up to (ℓ,m)=(6,6). This study paves the way for more realistic and detailed studies of compact objects and stellar collapse in full three dimensions and in large computational domains. The multipatch infrastructure and the improvements to mesh refinement and hydrodynamics codes discussed in this paper will be made available as part of the open-source Einstein Toolkit

An Application of Gaussian Process Modeling for High-order Accurate Adaptive Mesh Refinement Prolongation

We present a new polynomial-free prolongation scheme for Adaptive Mesh Refinement (AMR) simulations of compressible and incompressible computational fluid dynamics. The new method is constructed using a multi-dimensional kernel-based Gaussian Process (GP) prolongation model. The formulation for this scheme was inspired by the GP methods introduced by A. Reyes et al. (A New Class of High-Order Methods for Fluid Dynamics Simulation using Gaussian Process Modeling, Journal of Scientific Computing, 76 (2017), 443-480; A variable high-order shock-capturing finite difference method with GP-WENO, Journal of Computational Physics, 381 (2019), 189-217). In this paper, we extend the previous GP interpolations and reconstructions to a new GP-based AMR prolongation method that delivers a high-order accurate prolongation of data from coarse to fine grids on AMR grid hierarchies. In compressible flow simulations special care is necessary to handle shocks and discontinuities in a stable manner. To meet this, we utilize the shock handling strategy using the GP-based smoothness indicators developed in the previous GP work by A. Reyes et al. We demonstrate the efficacy of the GP-AMR method in a series of testsuite problems using the AMReX library, in which the GP-AMR method has been implemented

Gaussian Process Modeling for Upsampling Algorithms With Applications in Computer Vision and Computational Fluid Dynamics

Across a variety of fields, interpolation algorithms have been used to upsample lowresolution or coarse data fields. In this work, novel Gaussian Process based methodsare employed to solve a variety of upsampling problems. Specifically threeapplications are explored: coarse data prolongation in Adaptive Mesh Refinement(AMR) in the field of Computational Fluid Dynamics, accurate document imageupsampling to enhance Optical Character Recognition (OCR) accuracy, and fastand accurate Single Image Super Resolution (SISR). For AMR, a new, efficient,and “3rd order accurate” algorithm called GP-AMR is presented. Next, a novel,non-zero mean, windowed GP model is generated to upsample low resolution documentimages to generate a higher OCR accuracy, when compared to the industrystandard. Finally, a hybrid GP convolutional neural network algorithm is used togenerate a computationally efficient and high quality SISR model