1,494 research outputs found
Numerical integration of ordinary differential equations of various orders
Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced
Equation-free Dynamic Renormalization of a KPZ-type Equation
In the context of equation-free computation, we devise and implement a
procedure for using short-time direct simulations of a KPZ type equation to
calculate the self-similar solution for its ensemble averaged correlation
function. The method involves "lifting" from candidate pair-correlation
functions to consistent realization ensembles, short bursts of KPZ-type
evolution, and appropriate rescaling of the resulting averaged pair correlation
functions. Both the self-similar shapes and their similarity exponents are
obtained at a computational cost significantly reduced to that required to
reach saturation in such systems
Analysis of the accuracy and convergence of equation-free projection to a slow manifold
In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis, and A. Zagaris, Projecting to a
Slow Manifold: Singularly Perturbed Systems and Legacy Codes, SIAM J. Appl.
Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms
within the context of equation-free methods to approximate low-dimensional,
attracting, slow manifolds in systems of differential equations with multiple
time scales. For user-specified values of a finite number of the observables,
the m-th member of the class of algorithms (m = 0, 1, ...) finds iteratively an
approximation of the appropriate zero of the (m+1)-st time derivative of the
remaining variables and uses this root to approximate the location of the point
on the slow manifold corresponding to these values of the observables. This
article is the first of two articles in which the accuracy and convergence of
the iterative algorithms are analyzed. Here, we work directly with explicit
fast--slow systems, in which there is an explicit small parameter, epsilon,
measuring the separation of time scales. We show that, for each m = 0, 1, ...,
the fixed point of the iterative algorithm approximates the slow manifold up to
and including terms of O(epsilon^m). Moreover, for each m, we identify
explicitly the conditions under which the m-th iterative algorithm converges to
this fixed point. Finally, we show that when the iteration is unstable (or
converges slowly) it may be stabilized (or its convergence may be accelerated)
by application of the Recursive Projection Method. Alternatively, the
Newton-Krylov Generalized Minimal Residual Method may be used. In the
subsequent article, we will consider the accuracy and convergence of the
iterative algorithms for a broader class of systems-in which there need not be
an explicit small parameter-to which the algorithms also apply
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