24 research outputs found
Semi-implicit Krylov deferred correction algorithms, applications, and parallelization
In this dissertation, we introduce several strategies to improve the efficiency of the Krylov deferred correction (KDC) methods for special structured ordinary and partial differential equations with algebraic constraints. We first study the semi-implicit KDC (SI-KDC) technique which splits stiff differential equation systems into different components and applies different low-order time marching schemes to these components. Compared with the fully implicit KDC (FI-KDC) method, our analysis and preliminary numerical results for differential algebraic equations show that the SI-KDC schemes are more efficient due to the reduced number of operations in each spectral deferred correction (SDC) iteration. Next, we apply the SI-KDC scheme to simulate a two-scale model describing the mass transfer processes in drinking water treatment applications, in which some set of chemical species move from one distinct phase to a second distinct phase. We also present an improved effective model to further advance the efficiency of the multiscale modeling. Finally, we investigate the parareal method to parallelize the KDC techniques, and present some preliminary numerical results to show its potential in large scale simulations
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks
Large-scale multiphysics simulations are computationally challenging due to
the coupling of multiple processes with widely disparate time scales. The
advent of exascale computing systems exacerbates these challenges, since these
enable ever increasing size and complexity. Recently, there has been renewed
interest in developing multirate methods as a means to handle the large range
of time scales, as these methods may afford greater accuracy and efficiency
than more traditional approaches of using IMEX and low-order operator splitting
schemes. However, there have been few performance studies that compare
different classes of multirate integrators on complex application problems. We
study the performance of several newly developed multirate infinitesimal (MRI)
methods, implemented in the SUNDIALS solver package, on two reacting flow model
problems built on structured mesh frameworks. The first model revisits the work
of Emmet et al. (2014) on a compressible reacting flow problem with complex
chemistry that is implemented using BoxLib but where we now include comparisons
between a new explicit MRI scheme with the multirate spectral deferred
correction (SDC) methods in the original paper. The second problem uses the
same complex chemistry as the first problem, combined with a simplified flow
model, but run at a large spatial scale where explicit methods become
infeasible due to stability constraints. Two recently developed
implicit-explicit MRI multirate methods are tested. These methods rely on
advanced features of the AMReX framework on which the model is built, such as
multilevel grids and multilevel preconditioners. The results from these two
problems show that MRI multirate methods can offer significant performance
benefits on complex multiphysics application problems and that these methods
may be combined with advanced spatial discretization to compound the advantages
of both
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SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included