2,731 research outputs found
Peer Methods for the Solution of Large-Scale Differential Matrix Equations
We consider the application of implicit and linearly implicit
(Rosenbrock-type) peer methods to matrix-valued ordinary differential
equations. In particular the differential Riccati equation (DRE) is
investigated. For the Rosenbrock-type schemes, a reformulation capable of
avoiding a number of Jacobian applications is developed that, in the autonomous
case, reduces the computational complexity of the algorithms. Dealing with
large-scale problems, an efficient implementation based on low-rank symmetric
indefinite factorizations is presented. The performance of both peer approaches
up to order 4 is compared to existing implicit time integration schemes for
matrix-valued differential equations.Comment: 29 pages, 2 figures (including 6 subfigures each), 3 tables,
Corrected typo
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
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations
We are concerned with the efficient implementation of symplectic implicit
Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian)
ordinary differential equations by means of Newton-like iterations. We pay
particular attention to symmetric symplectic IRK schemes (such as collocation
methods with Gaussian nodes). For a -stage IRK scheme used to integrate a
-dimensional system of ordinary differential equations, the application of
simplified versions of Newton iterations requires solving at each step several
linear systems (one per iteration) with the same real
coefficient matrix. We propose rewriting such -dimensional linear systems
as an equivalent -dimensional systems that can be solved by performing
the LU decompositions of real matrices of size . We
present a C implementation (based on Newton-like iterations) of Runge-Kutta
collocation methods with Gaussian nodes that make use of such a rewriting of
the linear system and that takes special care in reducing the effect of
round-off errors. We report some numerical experiments that demonstrate the
reduced round-off error propagation of our implementation
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