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1.
S. Yu. Dobrokhotov 《Theoretical and Mathematical Physics》1997,112(1):827-843
According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only
three types of singularities that are in general position and have the property of “structure self-similarity and stability.”
Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is
described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions
for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations,
we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential.
This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result
can be used to predict the trajectory of the vortex center if we know its observable part.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66. 相似文献
2.
To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is
very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard
Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize.
We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with
respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the
estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori
step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems
of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and
nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision,
while the aggregative Taylor series method achieves the best computational time.
The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of
ODEs describing the optimal flight of a spacecraft from the Earth to the Moon.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical
Systems and Optimization, 2005. 相似文献
3.
The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated
as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution
at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation
of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence
the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it
can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled
systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which
enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic
fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities”
argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes
into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP
scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the
“acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach
of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization
of the Godunov scheme. 相似文献
4.
We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton
method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in
the context of the numerical solution of PDEs) suffer from a “loss of derivatives”, and that the subsequent linear rate of
convergence can be improved to be superlinear using an adaptation of Nash–Moser iteration for numerical analysis purposes;
the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these
ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate
the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also
in the inversion step.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
5.
In the first part of the present paper, we established estimates for the rate of approach of the integrals of a family of
“physical” white noises to a family of Wiener processes. We use this result to establish the estimate for the rate of approach
of a family of solutions of ordinary differential equations perturbed by some “physical” white noises to a family of solutions
of the corresponding It? equations. We consider both the case where the coefficient of random perturbation is separated from
zero and the case where it is not separated from zero. 相似文献
6.
Within the research into some geothermal modes, a 3D heat transfer process was described by a first-order system of differential equations (in terms of “temperature-heat-flow”). This system was solved by an explicit scheme for the mixed finite element spatial approximations based on the Raviart-Thomas degrees of freedom. In this paper, several algorithms based on the splitting technique for the vector heat-flow equation are proposed. Some comparison results of accuracy of the algorithms proposed are presented. 相似文献
7.
Partha Niyogi Stephen Smale Shmuel Weinberger 《Discrete and Computational Geometry》2008,39(1-3):419-441
Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces.
We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold
of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to
do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits
the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data
are “noisy” and lie near rather than on the submanifold in question.
The main results of this paper were first presented at a conference in honor of John Franks and Clark Robinson at Northwestern
University in April 2003. These results were formally written as Technical Report No. TR-2004-08, Department of Computer Science,
University of Chicago. 相似文献
8.
It has been shown [17,18,21] that the notion of index for DAEs (Differential Algebraic Equations), or more generally implicit
differential equations, could be interpreted in the framework of the formal theory of PDEs. Such an approach has at least
two decisive advantages: on the one hand, its definition is not restricted to a “state-space” formulation (order one systems),
so that it may be computed on “natural” model equations coming from physics (which can be, for example, second or fourth order
in mechanics, second order in electricity, etc.) and there is no need to destroy this natural way through a first order rewriting.
On the other hand, this formal framework allows for a straightforward generalization of the index to the case of PDEs (either
“ordinary” or “algebraic”). In the present work, we analyze several notions of index that appeared in the literature and give
a simple interpretation of each of them in the same general framework and exhibit the links they have with each other, from
the formal point of view. Namely, we shall revisit the notions of differential, perturbation, local, global indices and try
to give some clarification on the solvability of DAEs, with examples on time-varying implicit linear DAEs. No algorithmic
results will be given here (see [34,35] for computational issues) but it has to be said that the complexity of computing the
index, whatever approach is taken, is that of differential elimination, which makes it a difficult problem. We show that in
fact one essential concept for our approach is that of formal integrability for usual DAEs and that of involution for PDEs.
We concentrate here on the first, for the sake of simplicity. Last, because of the huge amount of work on DAEs in the past
two decades, we shall mainly mention the most recent results.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
9.
In this paper we present a technique to study the existence of rational solutions for systems of differential equations —
for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality
characterisation that involves matrix Padé approximants. It is important to note that, when the solution is rational, we use
formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We
expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main
ideas are applied in numerical resolution of partial differential equations.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
10.
We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations — sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse — the “fundamental theorem” — that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion. 相似文献
11.
Eduardo A. Notte-Cuello Waldyr A. Rodrigues Jr. 《Advances in Applied Clifford Algebras》2009,19(1):113-145
We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle
over a Lorentzian manifold, the true meaning of Freud’s identity of differential geometry discovered in 1939 (as a generalization
of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that
contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud’s identity (which
is a decomposition of the Einstein indexed 3-forms in two gauge dependent objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way
that Freud’s identity is usually applied in the formulation of energy-momentum “conservation laws” in GR. In order for this
paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms)
all calculations are done with all details (disclosing some of the “tricks of the trade” of the subject).
相似文献
12.
N. Ya. Moiseev 《Computational Mathematics and Mathematical Physics》2011,51(5):862-875
An approach to the construction of high-order accurate implicit predictor-corrector schemes is proposed. The accuracy is improved
by choosing a special time integration step for computing numerical fluxes through cell interfaces by using an unconditionally
stable implicit scheme. For smooth solutions of advection equations with constant coefficients, the scheme is second-order
accurate. Implicit difference schemes for multidimensional advection equations are constructed on the basis of Godunov’s method
with splitting over spatial variables as applied to the computation of “large” values at an intermediate layer. The numerical
solutions obtained for advection equations and the radiative transfer equations in a vacuum are compared with their exact
solutions. The comparison results confirm that the approach is efficient and that the accuracy of the implicit predictor-corrector
schemes is improved. 相似文献
13.
Matthias Aschenbrenner Lou van den Dries Joris van der Hoeven 《Selecta Mathematica, New Series》2005,11(2):247-280
H-fields are ordered differential fields that capture some basic properties of Hardy fields and fields of transseries. Each
H-field is equipped with a convex valuation, and solving first-order linear differential equations in H-field extensions is strongly affected by the presence of a “gap” in the value group. We construct a real closed H-field that solves every first-order linear differential equation, and that has a differentially algebraic H-field extension with a gap. This answers a question raised in [1]. The key is a combinatorial fact about the support of transseries
obtained from iterated logarithms by algebraic operations, integration, and exponentiation. 相似文献
14.
Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting
of a surjective linear map (“differential operator”) and an orthogonally closed subspace of the dual space (“boundary conditions”).
Defining the composition of boundary problems corresponding to their Green’s operators in reverse order, we characterize and
construct all factorizations of a boundary problem from a given factorization of the defining operator. For the case of ordinary
differential equations, the main results can be made algorithmic. We conclude with a factorization of a boundary problem for
the wave equation.
This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322. 相似文献
15.
Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems
Vicente Costanza Pablo S. Rivadeneira Ruben D. Spies 《Journal of Optimization Theory and Applications》2011,149(1):26-46
Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular
optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s
canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations
(PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems
with ℝ
n
-valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial
values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic
regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes.
The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to
reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems.
Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes
of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against
those obtained by using shooting techniques. 相似文献
16.
In this paper a new class of proximal-like algorithms for solving monotone inclusions of the form T(x)∋0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in order to solve the
differential inclusion , which can be viewed as a generalization of the steepest decent method for a convex function. It is proved that under suitable
conditions on the parameters of the LMM, the generated sequence converges weakly to a point in the solution set T
−1
(0). The LMM is very similar to the classical proximal point algorithm in that both are based on approximately evaluating
the resolvants of T. Consequently, LMM can be used to derive multi-step versions of many of the optimization methods based on the classical proximal
point algorithm. The convergence analysis allows errors in the computation of the iterates, and two different error criteria
are analyzed, namely, the classical scheme with summable errors, and a recently proposed more constructive criterion.
Received: April 2001 / Accepted: November 2002
Published online: February 14, 2003
Key Words. proximal point algorithm – monotone operator – numerical integration – strong stability – relative error criterion
Mathematics Subject Classification (1991): 20E28, 20G40, 20C20 相似文献
17.
In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects "virtual manifolds/orbifolds', on which we develop the integration theory. In particular, the virtual localization formula is obtained. 相似文献
18.
19.
A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual
problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as “dual” to the Dinkelbach-type
algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem
from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is
also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution
of the primal problem. We also consider a variant of this new algorithm, based on scaling the “dual” parametric function.
The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm
and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears
that contrary to the primal approach, the “dual” approach is less influenced by scaling.
This research was carried out at the Econometric Institute, Erasmus University, Rotterdam, the Netherlands and was supported
by J.N.I.C.T. (Portugal) under contract BD/707/90-RM. 相似文献
20.
We analyze results concerning the application of the numerical-analytic method suggested by Samoilenko to delay differential
equations, differential equations with “maxima,” functional-differential, operator-differential, and integro-differential
equations.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 960–979, July, 1998. 相似文献