20,147 research outputs found
An almost symmetric Strang splitting scheme for nonlinear evolution equations
In this paper we consider splitting methods for the time integration of
parabolic and certain classes of hyperbolic partial differential equations,
where one partial flow can not be computed exactly. Instead, we use a numerical
approximation based on the linearization of the vector field. This is of
interest in applications as it allows us to apply splitting methods to a wider
class of problems from the sciences.
However, in the situation described the classic Strang splitting scheme,
while still a method of second order, is not longer symmetric. This, in turn,
implies that the construction of higher order methods by composition is limited
to order three only. To remedy this situation, based on previous work in the
context of ordinary differential equations, we construct a class of Strang
splitting schemes that are symmetric up to a desired order.
We show rigorously that, under suitable assumptions on the nonlinearity,
these methods are of second order and can then be used to construct higher
order methods by composition. In addition, we illustrate the theoretical
results by conducting numerical experiments for the Brusselator system and the
KdV equation
Splitting methods in the numerical integration of non-autonomous dynamical systems
[EN] We present a procedure leading to efficient splitting schemes for the time integration of explicitly time dependent partitioned linear differential equations arising when certain partial differential equations are previously discretized in space. In the first stage we analyze the order conditions of the corresponding autonomous problem and construct new 6th-order methods. In the second stage, by following a procedure previously designed by the authors, we generalize the methods to the time dependent case in such a way that no order reduction is present. The resulting schemes compare favorably with other integrators previously available.This work has been supported by Ministerio de Ciencia e Innovacion (Spain) under project MTM2007-61572(co-financed by the ERDF of the European Union). SB also acknowledges financial support from Generalitat Valenciana through project GV/2009/032.Blanes Zamora, S.; Casas Perez, F.; Murua, A. (2012). Splitting methods in the numerical integration of non-autonomous dynamical systems. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 106(1):49-66. https://doi.org/10.1007/s13398-011-0024-849661061Blanes S., Casas F.: Splitting methods for non-autonomous separable dynamical systems. J. Phys. A. Math. Gen. 39, 5405–5423 (2006)Blanes S., Casas F., Murua A.: Symplectic splitting operator methods tailored for the time-dependent Schrödinger equation. J. Chem. Phys. 124, 234105 (2006)Blanes S., Casas F., Murua A.: Splitting methods for non-autonomous linear systems. Int. J. Comput. Math. 84, 713–727 (2007)Blanes S., Casas F., Murua A.: On the linear stability of splitting methods. Found. Comp. Math. 8, 357–393 (2008)Blanes S., Casas F., Murua A.: Splitting and composition methods in the numerical integration of differential equations. Bol. Soc. Esp. Math. Apl. 45, 87–143 (2008)Blanes, S., Casas, F., Murua, A.: Error analysis of splitting methods for the time dependent Schrödinger equation. arXiv:1001.1549 (2011)Blanes S., Casas F., Oteo J.A., Ros J.: The Magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009)Blanes S., Casas F., Ros J.: Improved high order integrators based on Magnus expansion. BIT 40, 434–450 (2000)Blanes S., Diele F., Marangi C., Ragni S.: Splitting and composition methods for explicit time dependence in separable dynamical systems. J. Comput. Appl. Math. 235, 646–659 (2010)Blanes S., Moan P.C.: Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 142, 313–330 (2002)Gray S., Manolopoulos D.E.: Symplectic integrators tailored to the time-dependent Schrödinger equation. J. Chem. Phys. 104, 7099–7112 (1996)Gray S., Verosky J.M.: Classical Hamiltonian structures in wave packet dynamics. J. Chem. Phys. 100, 5011–5022 (1994)Hairer E., Lubich C., Wanner G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd ed. Springer, Berlin (2006)Iserles A., Munthe-Kaas H.Z., Nørsett S.P., Zanna A.: Lie group methods. Acta Numer. 9, 215–365 (2000)Leimkuhler B., Reich S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004)Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)McLachlan R.I, Quispel R.: Splitting methods. Acta Numer. 11, 341–434 (2002)McLachlan R.I, Quispel R.G.W.: Geometric integrators for ODEs. J. Phys. A. Math. Gen. 39, 5251–5285 (2006)Rieben R., White D., Rodrigue G.: High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations. IEEE Trans. Antennas Propag. 52, 2190–2195 (2004)Sanz-Serna J.M., Calvo M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)Sanz-Serna J.M., Portillo A.: Classical numerical integrators for wave-packet dynamics. J. Chem. Phys. 104, 2349–2355 (1996)Sofroniou M., Spaletta G.: Derivation of symmetric composition constants for symmetric integrators. Optim. Methods Softw. 20, 597–613 (2005)Walker R.B., Preston K.: Quantum versus classical dynamics in treatment of multiple photon excitation of anharmonic-oscillator. J. Chem. Phys. 67, 2017–2028 (1977
An almost symmetric Strang splitting scheme for the construction of high order composition methods
In this paper we consider splitting methods for nonlinear ordinary
differential equations in which one of the (partial) flows that results from
the splitting procedure can not be computed exactly. Instead, we insert a
well-chosen state into the corresponding nonlinearity ,
which results in a linear term whose exact flow can be
determined efficiently. Therefore, in the spirit of splitting methods, it is
still possible for the numerical simulation to satisfy certain properties of
the exact flow. However, Strang splitting is no longer symmetric (even though
it is still a second order method) and thus high order composition methods are
not easily attainable. We will show that an iterated Strang splitting scheme
can be constructed which yields a method that is symmetric up to a given order.
This method can then be used to attain high order composition schemes. We will
illustrate our theoretical results, up to order six, by conducting numerical
experiments for a charged particle in an inhomogeneous electric field, a
post-Newtonian computation in celestial mechanics, and a nonlinear population
model and show that the methods constructed yield superior efficiency as
compared to Strang splitting. For the first example we also perform a
comparison with the standard fourth order Runge--Kutta methods and find
significant gains in efficiency as well better conservation properties
Splitting and composition methods in the numerical integration of differential equations
We provide a comprehensive survey of splitting and composition methods for
the numerical integration of ordinary differential equations (ODEs). Splitting
methods constitute an appropriate choice when the vector field associated with
the ODE can be decomposed into several pieces and each of them is integrable.
This class of integrators are explicit, simple to implement and preserve
structural properties of the system. In consequence, they are specially useful
in geometric numerical integration. In addition, the numerical solution
obtained by splitting schemes can be seen as the exact solution to a perturbed
system of ODEs possessing the same geometric properties as the original system.
This backward error interpretation has direct implications for the qualitative
behavior of the numerical solution as well as for the error propagation along
time. Closely connected with splitting integrators are composition methods. We
analyze the order conditions required by a method to achieve a given order and
summarize the different families of schemes one can find in the literature.
Finally, we illustrate the main features of splitting and composition methods
on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
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
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