Preconditioner updates applied to CFD model problems

Abstract

This paper deals with solving sequences of nonsymmetric linear systems with a block structure arising from compressible flow problems. The systems are solved by a preconditioned iterative method. We attempt to improve the overall solution process by sharing a part of the computational effort throughout the sequence. Our approach is fully algebraic and it is based on updating preconditioners by a block triangular update. A particular update is computed in a black-box fashion from the known preconditioner of some of the previous matrices, and from the difference of involved matrices. Results of our test compressible flow problems show, that the strategy speeds up the entire computation. The acceleration is particularly important in phases of instationary behavior where we saved about half of the computational time in the supersonic and moderate Mach number cases. In the low Mach number case the updated decompositions were similarly effective as the frozen preconditioners.

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

“…Our investigations within the finite element framework, see [29], show that if there is no additional global Newton-iteration, this procedure is more efficient because the number of unknown internal variables n Q is frequently larger than the unknown nodal displacements n u . However, one can not exclude circumstances where one or several additional global iterations might occur due to the "bad" estimation of the global coefficient matrix in (10) by the term coming from (9). In such cases the overall solution time is uncertain and can be larger than the original version of Table 1.…”

Preconditioner updates for solving sequences of linear systems in matrix-free environment

“…The simple updates (2.2) can be expected to be efficient when the dominant information in the difference matrix B is contained in one triangular part, like for instance with upwind/downwind finite difference discretizations. However, we showed in [1,33,34] that the updates, possibly combined with improvements like specific reordering, Gauss-Jordan transforms or Gauss-Seidel-type extensions, are beneficial for a much broader spectrum of problems (including some CFD simulations discretized with finite volumes or elements). One remarkable feature is that the updates seem to yield powerful preconditioning not only, as one would expect from the definition (2.2), when the system matrices are changing slowly.…”

“…One remarkable feature is that the updates seem to yield powerful preconditioning not only, as one would expect from the definition (2.2), when the system matrices are changing slowly. In [34] the updates are most efficient in the transient phase of the simulation where turbulence causes large differences between system matrices. This may be explained by the fact that we take into account part of the large differences through the matrix B in (2.2).…”

“…The authors in [31] propose approximate diagonal updates to solve parabolic PDEs, see also [32]. Nonsymmetric updates of general incomplete LU decompositions were considered in [1,33], see also some results in solving real-world problems in [34]. So far, neither of these approaches have addressed the challenges related to preconditioner updates in matrix-free environment.…”

Choosing order of operations to accelerate strip structure analysis in parameter range

“…The fourth approach is based on periodic recomputation of a preconditioner. The fifth approach is equivalent to the previous one, however, if, during some period, the number of iterations required to solve the current system is bigger than the sum of iterations required to solve the first system in this period and a predetermined threshold, then the preconditioner is updated [5]. The last approach is based on the adaptive use of information about Krylov subspaces obtained in the previous steps [6].…”