Grid Generation Techniques in Magnetic Reconnection Simulation

by Yuan, Xuefei, Ph.D., COLUMBIA UNIVERSITY, 2011, 291 pages; 3451820


Adaptive solution refinement through grid motion, with high solution variation regions serving as attractors for grid density or magnetic flux surfaces serving as constant-coordinate alignment targets, is a capability of increasing interest to continuum-based models of fusion devices and represents an advantage for numerical simulations.

The four-field extended MagnetoHydroDynamics (MHD) equations with hyper-resistivity terms are derived from a set of two-dimensional basic MHD equations describing incompressible, two-fluid (ion and electron), quasi-neutral plasma. However, numerical simulations present a difficult challenge because of demanding spatial resolution requirements. In particular, the out-of-plane current density can develop large gradients and near singular behavior in the reconnection region as time evolves, requiring localized regions of higher resolution.

A variational grid generation technique based on an equidistribution principle gives a single Monge-Ampère (MA) grid equation through Monge-Kantorovich (MK) optimization process with a constraint, scalar equation (called the solvability condition) on a density function. This method is applied to obtain adaptive grids for use in solving the MHD equations by designing this density function to be a function of the current density computed from solving the MHD equations. The grid generator defines a new grid at each time step, adaptive to the corresponding current density. As time advances, these grids track changes of the solution and provide higher resolution where it is needed. The absolute value of the current density raised to some power m and the ratio between the maximum cell size and the minimum cell size r enter into the determination of this density function. It is important that some smoothing is performed on the current density before inserting it into the density function, otherwise the overall solution is strongly sensitive to the values of m and r. The resulting structured, boundary-fitted coordinate grids allow a simple ordering of nodes and determination of neighbors, and thus are well-suited for algorithms that require decomposition for parallelization and regular low-overhead addressing of floating-point operands for high efficiency on cache-based and emerging heterogeneous processors.

The physical equations are transformed from Cartesian coordinates so that the solution-defined curvilinear coordinates replace Cartesian coordinates as the independent variables. The time-dependent sequence of grid coordinates defines a “grid velocity” that can be thought of as a Lagrangian velocity. It enters into the nonlinear system of equations so that it is not necessary to perform a separate interpolation step to transfer the solution from the grid at the nth time step to the grid at the ( n + 1)th time step. The transformed MHD equations are of the same type as the original ones, but are more complicated in the sense of having variable coefficients, cross derivatives, and more geometrical terms.

The application of an implicit scheme to the time-dependent problem prevents the time step size from being restricted by a Courant-Friedrichs-Lewy (CFL) condition, but only by accuracy. The Newton-Krylov-Schwarz (NKS) algorithm is used to solve above systems at each time iteration in parallel. Inexact Newton methods are applied as nonlinear solvers that iterate to a solution through a sequence of linear problems (Newton update equations) from an initial guess of the solution; Krylov subspace methods, such as Generalized Minimal RESidual (GMRES), are used as linear solvers to solve such Newton equations; and Schwarz methods are applied as preconditioners for the linear system.

We solve the four-field extended MHD equations in both Cartesian and curvilinear coordinates for a problem in which a localized reconnection region forms. Curvilinear solutions are compared with Cartesian solutions using bicubic interpolations to evaluate Cartesian solutions in curvilinear coordinates. The solution of physical problems in the statically adaptive curvilinear coordinate system improves upon that in the Cartesian coordinate system in accuracy, convergence, and usage of computer sources; moreover, dynamically adaptive curvilinear coordinate systems surpass the statically adaptive curvilinear coordinate system as time advances. Five global quantities of the system are compared in Cartesian solutions and curvilinear solutions with regard to convergence and accuracy. Convergence studies show that curvilinear solutions converge faster than Cartesian solutions, and accuracy studies show that curvilinear solutions can achieve the same accuracy as Cartesian solutions with fewer grid points. The ratio between grid points for Cartesian solutions and curvilinear solutions for the same accuracy is dependent on r, the ratio between the largest cell size and the smallest cell size in the density function of the grid generation technique. This is extremely helpful when the problem size is very large as the time step size is limited only by accuracy, not by the smallest grid size. The curvilinear system requires about twice the running time, memory, and flops of the Cartesian system, per grid point per iteration, but overall produces significant savings for the same accuracy.

Source TypeDissertation
SubjectsApplied mathematics
Publication Number3451820

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