We present a family of spacetree-based multigrid realizations using the tree's multiscale nature to derive coarse grids. They align with matrix-free geometric multigrid solvers as they never assemble the system matrices which is cumbersome for dynamically adaptive grids and full multigrid. The most sophisticated realizations use BoxMG to construct operator-dependent prolongation and restriction in combination with Galerkin/Petrov-Galerkin coarse-grid operators. This yields robust solvers for nontrivial elliptic problems. We embed the algebraic, problem-and grid-dependent multigrid operators as stencils into the grid and evaluate all matrix-vector products in-situ throughout the grid traversals. While such an approach is not literally matrix-free-the grid carries the matrix-we propose to switch to a hierarchical representation of all operators. Only differences of algebraic operators to their geometric counterparts are held. These hierarchical differences can be stored and exchanged with small memory footprint. Our realizations support arbitrary dynamically adaptive grids while they vertically integrate the multilevel operations through spacetree linearization. This yields good memory access characteristics, while standard colouring of mesh entities with domain decomposition allows us to use parallel manycore clusters. All realization ingredients are detailed such that they can be used by other codes.
In this paper, we develop a new technique for driving global non-potential simulations of the Sun's coronal magnetic field solely from sequences of radial magnetic maps of the solar photosphere. A primary challenge to driving such global simulations is that the required horizontal electric field cannot be uniquely determined from such maps. We show that an "inductive" electric field solution similar to that used by previous authors successfully reproduces specific features of the coronal field evolution in both single and multiple bipole simulations. For these cases, the true solution is known because the electric field was generated from a surface flux-transport model. The match for these cases is further improved by including the non-inductive electric field contribution from surface differential rotation. Then, using this reconstruction method for the electric field, we show that a coronal nonpotential simulation can be successfully driven from a sequence of ADAPT maps of the photospheric radial field, without including additional physical observations which are not routinely available.
SUMMARY The classical Petrov–Galerkin approach to Black Box multigrid for nonsymmetric problems due to Dendy is combined with the recent factor‐three‐coarsening Black Box algorithm due to Dendy and Moulton, along with a powerful symmetric line Gauss–Seidel smoother, resulting in an efficient and robust multigrid solver. Focusing on the convection–diffusion operator, the algorithm is tested and shown to achieve fast and reliable convergence with both first‐order and second‐order accurate upstream discretizations of the convection operator. The solver also exhibits robust behavior with respect to discontinuous jumps in the diffusion coefficient and performs well for recirculating flows over a wide range of diffusion coefficients. The efficiency of the solver is supported by results of an analysis for the case of constant coefficients. Copyright © 2012 John Wiley & Sons, Ltd.
Predictions of the physical parameters of the solar wind at Earth are at the core of operational space weather forecasts. Such predictions typically use line‐of‐sight observations of the photospheric magnetic field to drive a heliospheric model. The models Wang‐Sheeley‐Arge (WSA) and ENLIL for the transport in the heliosphere are commonly used for these respective tasks. Here we analyze the impact of replacing the potential field coronal boundary conditions from WSA with two alternative approaches. The first approach uses a more realistic nonpotential rather than potential approach, based on the Durham Magneto Frictional Code (DUMFRIC) model. In the second approach the ENLIL inner boundary conditions are based on Inter Planetary Scintillation observations (IPS). We compare predicted solar wind speed, plasma density, and magnetic field magnitude with observations from the WIND spacecraft for two 6‐month intervals in 2014 and 2016. Results show that all models tested produce fairly similar output when compared to the observed time series. This is not only reflected in fairly low correlation coefficients (<0.3) but also large biases. For example, for solar wind speed some models have average biases of more than 150 km/s. On a positive note, the choice of coronal magnetic field model has a clear influence on the model results when compared to the other models in this study. Simulations driven by IPS data have a high success rate with regard to detection of the high speed solar wind. Our results also indicate that model forecasts do not degrade for longer forecast times.
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