3,987 research outputs found
Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y)
In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y‴=f(x,y)y‴=f(x,y) denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge–Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs
An Algebraic Framework for the Real-Time Solution of Inverse Problems on Embedded Systems
This article presents a new approach to the real-time solution of inverse
problems on embedded systems. The class of problems addressed corresponds to
ordinary differential equations (ODEs) with generalized linear constraints,
whereby the data from an array of sensors forms the forcing function. The
solution of the equation is formulated as a least squares (LS) problem with
linear constraints. The LS approach makes the method suitable for the explicit
solution of inverse problems where the forcing function is perturbed by noise.
The algebraic computation is partitioned into a initial preparatory step, which
precomputes the matrices required for the run-time computation; and the cyclic
run-time computation, which is repeated with each acquisition of sensor data.
The cyclic computation consists of a single matrix-vector multiplication, in
this manner computation complexity is known a-priori, fulfilling the definition
of a real-time computation. Numerical testing of the new method is presented on
perturbed as well as unperturbed problems; the results are compared with known
analytic solutions and solutions acquired from state-of-the-art implicit
solvers. The solution is implemented with model based design and uses only
fundamental linear algebra; consequently, this approach supports automatic code
generation for deployment on embedded systems. The targeting concept was tested
via software- and processor-in-the-loop verification on two systems with
different processor architectures. Finally, the method was tested on a
laboratory prototype with real measurement data for the monitoring of flexible
structures. The problem solved is: the real-time overconstrained reconstruction
of a curve from measured gradients. Such systems are commonly encountered in
the monitoring of structures and/or ground subsidence.Comment: 24 pages, journal articl
Adaptive propagation of quantum few-body systems with time-dependent Hamiltonians
In this study, a variety of methods are tested and compared for the numerical
solution of the Schr\"odinger equation for few-body systems with explicitely
time-dependent Hamiltonians, with the aim to find the optimal one. The
configuration interaction method, generally applied to find stationary
eigenstates accurately and without approximations to the wavefunction's
structure, may also be used for the time-evolution, which results in a large
linear system of ordinary differential equations. The large basis sizes
typically present when the configuration interaction method is used calls for
efficient methods for the time evolution. Apart from efficiency, adaptivity (in
the time domain) is the other main focus in this study, such that the time step
is adjusted automatically given some requested accuracy. A method is suggested
here, based on an exponential integrator approach, combined with different ways
to implement the adaptivity, which was found to be faster than a broad variety
of other methods that were also considered.Comment: 16 pages, 1 figure (4 panels
Robust Exponential Runge-Kutta Embedded Pairs
Exponential integrators are explicit methods for solving ordinary
differential equations that treat linear behaviour exactly. The stiff-order
conditions for exponential integrators derived in a Banach space framework by
Hochbruck and Ostermann are solved symbolically by expressing the Runge--Kutta
weights as unknown linear combinations of phi functions. Of particular interest
are embedded exponential pairs that efficiently generate both a high- and
low-order estimate, allowing for dynamic adjustment of the time step. A key
requirement is that the pair be robust: if the nonlinear source function has
nonzero total time derivatives, the order of the low-order estimate should
never exceed its design value. Robust exponential Runge--Kutta (3,2) and (4,3)
embedded pairs that are well-suited to initial value problems with a dominant
linearity are constructed.Comment: 24 pages, 8 figures. The Mathematica scripts mentioned in the paper
can be found in: https://github.com/stiffode/expint
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