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399
Creating EvenlySpaced Streamlines of Arbitrary Density
, 1997
"... . This paper presents a new evenlyspaced streamlines placement algorithm to visualize 2D steady flows. The main technical contribution of this work is to propose a single method to compute a wide variety of flow field images, ranging from texturelike to handdrawing styles. Indeed the control ..."
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Cited by 120 (3 self)
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. This paper presents a new evenlyspaced streamlines placement algorithm to visualize 2D steady flows. The main technical contribution of this work is to propose a single method to compute a wide variety of flow field images, ranging from texturelike to handdrawing styles. Indeed the control of the density of the field is very easy since the user only needs to set the separating distance between adjacent streamlines, which is related to the overall density of the image. We show that our method produces images of a quality at least as good as other methods but that it is computationally less expensive and offers a better control on the rendering process. Introduction The problem of visualizing vector fields has been widely addressed in the past years because it has numerous applications. The main issue is to visualize properly the direction and magnitude of the flow. Spatial resolution techniques such as arrow plots, streamlines or particles traces suffer from their spatia...
Fast and Resolution Independent Line Integral Convolution
, 1995
"... Line Integral Convolution (LIC) is a powerful technique for generating striking images and animations from vector data. Introduced in 1993, the method has rapidly found many application areas, ranging from computer arts to scientific visualization. Based upon locally filtering an input texture along ..."
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Cited by 119 (9 self)
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Line Integral Convolution (LIC) is a powerful technique for generating striking images and animations from vector data. Introduced in 1993, the method has rapidly found many application areas, ranging from computer arts to scientific visualization. Based upon locally filtering an input texture along a curved stream line segment in a vector field, it is able to depict directional information at high spatial resolutions. We present a new method for computing LIC images. It employs simple box filter kernels only and minimizes the total number of stream lines to be computed. Thereby it reduces computational costs by an order of magnitude compared to the original algorithm. Our method utilizes fast, errorcontrolled numerical integrators. Decoupling the characteristic lengths in vector field grid, input texture and output image, it allows computation of filtered images at arbitrary resolution. This feature is of significance in computer animation as well as in scientific visualization, whe...
Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 96 (18 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
ImplicitExplicit RungeKutta Methods for TimeDependent Partial Differential Equations
 Appl. Numer. Math
, 1997
"... Implicitexplicit (IMEX) linear multistep timediscretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable timestep restrictions when applied to convectiondiffusion problems, unless diffusion strongly dominates and an ap ..."
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Cited by 87 (3 self)
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Implicitexplicit (IMEX) linear multistep timediscretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable timestep restrictions when applied to convectiondiffusion problems, unless diffusion strongly dominates and an appropriate BDFbased scheme is selected [2]. In this paper, we develop RungeKuttabased IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range. 1 Introduction When a timedependent partial differential equation (PDE) involves terms of different types, it is a natural idea to employ different discretizations for them. Implicitexplicit (IMEX) timediscretization schemes are an example of such a strategy. Linear multistep IMEX schemes have been used by many researchers, especially in conjunction with spectral methods [10, 3]. Some schemes of this type were proposed and analyzed as far back as the late 1970's [15, 5]. Instances of...
Backward Error Analysis for Numerical Integrators
 SIAM J. Numer. Anal
, 1996
"... We consider backward error analysis of numerical approximations to ordinary differential equations, i.e., the numerical solution is formally interpreted as the exact solution of a modified differential equation. A simple recursive definition of the modified equation is stated. This recursion is used ..."
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Cited by 69 (7 self)
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We consider backward error analysis of numerical approximations to ordinary differential equations, i.e., the numerical solution is formally interpreted as the exact solution of a modified differential equation. A simple recursive definition of the modified equation is stated. This recursion is used to give a new proof of the exponentially closeness of the numerical solutions and the solutions to an appropriate truncation of the modified equation. We also discuss qualitative properties of the modified equation and apply these results to the symplectic variable stepsize integration of Hamiltonian systems, the conservation of adiabatic invariants, and numerical chaos associated to homoclinic orbits. 3 S. Reich, Backward error analysis 4
The LifeSpan of Backward Error Analysis for Numerical Integrators
 Numer. Math
, 1996
"... this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation. Results on the longtime behaviour of numerical solutions are obtained in this way. We present applications to the numerical phase po ..."
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Cited by 64 (3 self)
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this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation. Results on the longtime behaviour of numerical solutions are obtained in this way. We present applications to the numerical phase portrait near hyperbolic equilibrium points, to asymptotically stable periodic orbits and Hopf bifurcation, and to energy conservation and approximation of invariant tori in Hamiltonian systems.
Backward Analysis Of Numerical Integrators And Symplectic Methods
 Stiff and DifferentialAlgebraic Problems
, 1994
"... . A backward analysis of integration methods, whose numerical solution is a Pseries, is presented. Such methods include RungeKutta methods, partitioned RungeKutta methods and Nystrom methods. It is shown that the numerical solution can formally be interpreted as the exact solution of a perturbed d ..."
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Cited by 52 (7 self)
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. A backward analysis of integration methods, whose numerical solution is a Pseries, is presented. Such methods include RungeKutta methods, partitioned RungeKutta methods and Nystrom methods. It is shown that the numerical solution can formally be interpreted as the exact solution of a perturbed differential system whose righthand side is again a Pseries. The main result of this article is that for symplectic integrators applied to Hamiltonian systems the perturbed differential equation is a Hamiltonian system too. The proofs use the onetoone correspondence between rooted trees and the expressions appearing in the Taylor expansions of the exact and numerical solutions (elementary differentials). Key words. Backward analysis, Hamiltonian systems, RungeKutta methods, symplectic methods, Pseries. 1. Introduction For the numerical solution of ordinary differential equations y 0 = f(y) (1:1) we consider onestep integrators such as RungeKutta methods or something similar. The ...
Computations in a Free Lie Algebra
, 1998
"... Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the BakerCampbellHausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity ..."
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Cited by 51 (16 self)
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Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the BakerCampbellHausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witts formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides accurate bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new...
Numerical solution of isospectral flows
 Math. of Comp
, 1997
"... Abstract. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L ′ =[B(L),L], L(0) = L0, where L0 is a d × d symmetric matrix, B(L) is a skewsymmetric matrix function of L and [B, L] is the Lie bracket ..."
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Cited by 50 (23 self)
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Abstract. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L ′ =[B(L),L], L(0) = L0, where L0 is a d × d symmetric matrix, B(L) is a skewsymmetric matrix function of L and [B, L] is the Lie bracket operator. We show that standard Runge–Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order. 1. Background and notation 1.1. Introduction. The interest in solving isospectral flows is motivated by their relevance in a wide range of applications, from molecular dynamics to micromagnetics to linear algebra. The general form of an isospectral flow is the differential
Chaos in Lorenz equations: a computer assisted proof. Part III: The classical case, in preparation
"... Abstract. Details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with rigorous computer assisted computations. As an application of these methods it ..."
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Cited by 46 (25 self)
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Abstract. Details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with rigorous computer assisted computations. As an application of these methods it is proven that for some explicit parameter values the Lorenz equations exhibit chaotic dynamics. 1.