Accelerated Modeling Of Three-dimensional Ambient Solar Wind

How can we forecast space weather a few hours to a few days in advance is not only an important important subject in the field of space weather,but also the significant demand of modern technological development and globalization for space weather forecasts.The threedimensional numerical study of solar wind coronal interplanetary processes,which is based on the MHD and the powerful computing capacity,is an important means of space weather forecast modeling.And the ambient solar wind is an important basis for studying disturbance propagation and forecasting space weather conditions.Numerically solving the governing equations of the ambient solar wind is not an easy work due to its complexity by combining the fluid dynamic equations with the Maxwells equations of electromagnetics,as well as the solenoidal constraint.Besides,the spherical shell computational domain with a vast spatial scale enlarges the difficulties of numerical computation.Moreover,near real-time simulation and forecast propose harsher requirements for computational efficiency.Considering these factors,we construct a new,concise,flexible and quick solar wind model.In this paper,we first construct hexahedral mesh elements and build a basic finite volume(FV)scheme on the six-component grid system,which is of sphere-surface body-fitting and can avoid the sigularity of polar grid.We use the Powell source term and the hyperbolic divergence cleaning methods to numerically satisfy the constraint of · B = 0,use rotation matrice to simplify the fomulas and codes,use Lax-Friedrichs,Roe etc.to get the approximate Riemann solver,and use the linear least squre method to acquire spatially sencond order accuracy.The numerical results of testcases and Carrington rotations validate that this model can not only obtain the desired spatial accuracy,but capture the basic features and structres of the solar wind accurately.Then based on this basic scheme,we study the accelerating method of the ambient solar wind simulation.In the respect of numerical computation methods,we use the generalized minimum residual(GMRES)algorithm with an lower–upper symmetric Gauss–Seidel(LU-SGS)preconditioner for the model.GMRES algorithm is a generalization of the conjugate gradient method,which minimizes the norm of the computed residual vector over the subspace spanned by a certain number of orthogonal search directions.LU-SGS acts as a preconditioner to cluster the eigenvalues at a single value,and can improve the efficiency and robustness of GMRES.This implicit method not only has a high efficiency of iterations,but also can completely eliminate the storage of the Jacobian matrix inherent in the original formulation by making some approximations to the implicit operator,which is very important to large,distributed computation.As the preconditioner will change the time iterm of the governing equations and make the physical time inaccurate,we also use the dual-time step in the implicit scheme.Compared with the basic scheme,the implicit scheme can enlarge the CFL number more than 100 times,reduce the computational time for the steady-state study from several days to only a few hours,and achieve speedup ratios of about 30 x.In the respect of computating devices,by utilizing GPU's advantage in intensive calculation,we adopt the techniques of MPI and CUDA,and implement our basic scheme on multiple GPUs.In order to explore the computing power of devices as much as possible,we take CUDA optimization strategy and load balance into considerations.Compared with the execution time that costs on same number of CPU processes,the multi-GPU scheme can acquire speedup ratios of nearly 30 x.