On the solution of evolution equations based on multigrid and explicit iterative methods

On Development of Parallel Algorithms for Solving Parabolic and Elliptic Equations

“…Let computations be performed on a processor grid (d-dimensional, d = 1, 2, 3) with some given number of processors. We assume that the computational grid of the discrete problem itself is many times larger than the specified number of processors, and grid equations on the roughest grid are distributed by processors in such a way that the number of nodes of the rough grid is not too small for parallel computing by using the iterative process (21). For example, the problem on the grid with the number of nodes 4096 3 satisfies this requirement when solving on a cubic processor grid with the number of processors from 16 3 to 64 3 at 5 grid levels in the case where each MM level is obtained from the previous level by deleting every second geometrical point of the grid at each direction.…”

“…We consider only the classical version of the multigrid method with resampling on all additional grid levels. Numerous meaningful examples of solving problems with various degrees of complexity (with discontinuous anisotropic coefficients and significant anisotropy) are given in [18][19][20][21]. For demonstration of the computational efficiency, we provide the results of the solution of the simplest equation u t = Δu + f in the parallelepiped…”

“…The amplitude of such components decreases almost inversely proportional to the frequency, which corresponds to the needs of the multigrid method. As a result, for the geometric classical multigrid method (MM) on Cartesian grids, we have constructed an effective parallel adaptive implementation (see [18][19][20][21]); now we develop it for the algebraic multigrid method.…”

Modelling the bow shock Pulsar Wind Nebulae propagating through a non-uniform ISM

“…We used an axisymmetric, non-relativistic resistive MHD code. The code incorporates the methods of local iterations (Zhukov, Zabrodin & Feodoritova 1993) and flux-corrected transport (Boris & Book 1973). The flow is described by the resistive MHD equations (Landau & Lifshitz 1960):…”

Coarse grid corrections in Krylov subspace evaluations of the matrix exponential

“…Fedorenko [11,12], the first papers describing the multigrid method as we know it now [21, Section 10.9.2], is devoted to the solution of Poisson equations arising in time integration of 2D incompressible hydrodynamics equations [13]. Currently, multigrid methods form a major tool for efficient implementation of implicit and semi-implicit time integration schemes on parallel supercomputers [2,17,36,35].…”