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29
Automatic Validation of Numerical Solutions
, 1997
"... this report . . . 73 7.2 Periodic solutions of autonomous systems . . . . . . . . . . . . 74 7.2.1 The Brusselator . . . . . . . . . . . . . . . . . . . . . . 75 7.2.2 The Lorenz system . . . . . . . . . . . . . . . . . . . . 76 7.2.3 The Van der Pol system . . . . . . . . . . . . . . . . . . 80 7.3 ..."
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Cited by 27 (1 self)
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this report . . . 73 7.2 Periodic solutions of autonomous systems . . . . . . . . . . . . 74 7.2.1 The Brusselator . . . . . . . . . . . . . . . . . . . . . . 75 7.2.2 The Lorenz system . . . . . . . . . . . . . . . . . . . . 76 7.2.3 The Van der Pol system . . . . . . . . . . . . . . . . . . 80 7.3 Periodic solutions of nonautonomous systems . . . . . . . . . . 82 7.3.1 The forced Brusselator . . . . . . . . . . . . . . . . . . 83 7.4 Solutions of boundary value problems . . . . . . . . . . . . . . 84
Numerical integrators that preserve symmetries and reversing symmetries
 SIAM J. Numer. Anal
, 1998
"... Abstract. We consider properties of flows, the relationships between them, and whether numerical integrators can be made to preserve these properties. This is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields. This new framewo ..."
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Cited by 21 (14 self)
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Abstract. We consider properties of flows, the relationships between them, and whether numerical integrators can be made to preserve these properties. This is done in the context of automorphisms and antiautomorphisms of a certain group generated by maps associated to vector fields. This new framework unifies several known constructions. We also use the concept of “covariance” of a numerical method with respect to a group of coordinate transformations. The main application is to explore the relationship between spatial symmetries, reversing symmetries, and time symmetry of flows and numerical integrators.
Attractive Invariant Manifolds Under Approximation Part I: Inertial Manifolds
 J. Diff. Eq
, 1995
"... A class of nonlinear dissipative partial differential equations that possess finite dimensional attractive invariant manifolds is considered. An existence and perturbation theory is developed which unifies the cases of unstable manifolds and inertial manifolds into a single framework. It is shown th ..."
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Cited by 12 (2 self)
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A class of nonlinear dissipative partial differential equations that possess finite dimensional attractive invariant manifolds is considered. An existence and perturbation theory is developed which unifies the cases of unstable manifolds and inertial manifolds into a single framework. It is shown that certain approximations of these equations, such as those arising from spectral or finite element methods in space, onestep timediscretization or a combination of both, also have attractive invariant manifolds. Convergence of the approximate manifolds to the true manifolds is established as the approximation is refined. In this part of the paper applications to the behavior of inertial manifolds under approximation are considered. From this analysis deductions about the structure of the attractor and the flow on the attractor under discretization can be made. 1 Introduction Attractive invariant manifolds for evolution equations are fundamental in understanding longtime dynamics. Import...
Nonlinear Hybrid Procedures and Fixed Point Iterations
, 1998
"... Let (x n ) and (x n ) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (y n ) with better convergence properties than (x n ) and (x n ). Then, this procedure is used for accelerating the conve ..."
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Cited by 11 (7 self)
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Let (x n ) and (x n ) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (y n ) with better convergence properties than (x n ) and (x n ). Then, this procedure is used for accelerating the convergence of a given sequence and applied to the construction of fixed point methods. New methods are derived. Finally, the connection between fixed point iterations and methods for the numerical integration of differential equations is also exploited. Numerical results are given.
ANALYSIS OF THE DYNAMICS OF LOCAL ERROR CONTROL VIA A PIECEWISE CONTINUOUS RESIDUAL
 BIT
, 1998
"... Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpo ..."
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Cited by 9 (4 self)
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Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under the assumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for the MATLAB ode23 algorithm [10] when applied to a variety of problems. Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the longtime dynamical properties of the equationdissipative, contractive and gradient systems are analysed in this way.
Combining trust region techniques and Rosenbrock methods for gradient systems
, 2006
"... Rosenbrock methods are popular for solving stiff initial value problems for ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas and implicit RungeKutta ..."
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Cited by 9 (5 self)
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Rosenbrock methods are popular for solving stiff initial value problems for ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas and implicit RungeKutta methods. In this paper, we introduce some trust region techniques to control the time step in the second order Rosenbrock methods for gradient systems. These techniques are different from the local error control schemes. Both the global and local convergence of the new class of trust region Rosenbrock methods for solving the equilibrium points of gradient systems are addressed. Finally some promising numerical results are presented.
Symplectic RungeKutta Schemes I: Order Conditions
 SIAM J. NUM. ANAL
, 1997
"... Much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating Hamiltonian systems of differential equations. RungeKutta schemes are symplectic subject to a simple algebraic condition. Starting with Butcher's formalism it is s ..."
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Cited by 6 (3 self)
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Much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating Hamiltonian systems of differential equations. RungeKutta schemes are symplectic subject to a simple algebraic condition. Starting with Butcher's formalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods; involving only s(s + 1)=2 free parameters for a symplectic s stage scheme. A graph theoretical process for determining the new order conditions is outlined. Furthermore, it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the parametrised coefficients. When coupled with the standard simplifying assumptions for implicit schemes the number of order conditions may be further reduced. In the new framework a simple symmetry of the parameter matrix yields (not necessarily symplectic) selfadjoint methods. In this case the order ...
Algorithms for Constructing Stable Manifolds of Stationary Solutions
"... Algorithms for computing stable manifolds of hyperbolic stationary solutions of autonomous systems are of two types: either the aim is to compute a single point on the manifold or the entire (local) manifold. Traditionally only indirect methods have been considered, i.e. first the continuous problem ..."
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Cited by 5 (1 self)
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Algorithms for computing stable manifolds of hyperbolic stationary solutions of autonomous systems are of two types: either the aim is to compute a single point on the manifold or the entire (local) manifold. Traditionally only indirect methods have been considered, i.e. first the continuous problem is discretised by a onestep scheme and then the LiapunovPerron or Hadamard graph transform are applied to the resulting discrete dynamical system. We will consider different variants of these indirect methods but also algorithms of the above two types which are applied directly to the continuous problem.
Numerical analysis of parabolic pLaplacian: approximation of trajectories
 Siam J. Numer. Anal
, 2000
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Convergence And Stability In The Numerical Approximation Of Dynamical Systems
"... this article we give an overview of the application of theories from dynamical systems to the analysis of numerical methods for initialvalue problems. We start by describing the classical viewpoints of numerical analysis and of dynamical systems and then indicate how the two viewpoints can be merge ..."
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Cited by 4 (0 self)
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this article we give an overview of the application of theories from dynamical systems to the analysis of numerical methods for initialvalue problems. We start by describing the classical viewpoints of numerical analysis and of dynamical systems and then indicate how the two viewpoints can be merged to provide a framework for both the interpretation of data obtained from numerical simulations and the design of effecient numerical methods. This is done in section 2. In addressing the question of how to interpret data, we will show in section 3 how the concept of convergence can be generalized to the numerical approximation of dynamical systems. The main theory is developed for onestep methods for ordinary differential equations and extensions to the study of multistep methods, adaptive timestepping algorithms and partial differential equations are then outlined. In addressing the question of designing effecient schemes we will show in section 4 how the concept of stability can be generalized to dynamical systems. Stability theory is developed for both onestep and multistep methods for ordinary differential equations; extensions to adaptive timestepping and to partial differential equations are also outlined. A variety of surveys of this field already exist  see [13] for a complete study, see [9] for a discussion of convergence in the dynamical systems context, see [12] for a discussion of stability in the dynamical systems context and see [5] for a discussion of both issues in relation to discretization of partial differential equations. Since these surveys contain exhaustive bibliographys we refer to them for detailed references. 1