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Fourth order timestepping for low dispersion Kortewegde Vries and nonlinear Schrödinger equation
 27 T.P. Liu, Development of singularities in the
, 2008
"... Abstract. Purely dispersive equations, such as the Kortewegde Vries and the nonlinear Schrödinger equations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispers ..."
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Abstract. Purely dispersive equations, such as the Kortewegde Vries and the nonlinear Schrödinger equations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blowup. Fourth order timestepping in combination with spectral methods is beneficial to numerically resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kortewegde Vries and the focusing and defocusing nonlinear Schrödinger equations in the small dispersion limit: an exponential timedifferencing fourthorder RungeKutta method as proposed by Cox and Matthews in the implementation by Kassam and Trefethen, integrating factors, timesplitting, Fornberg and Driscoll’s ‘sliders’, and an ODE solver in Matlab.
Fiberdyne Systems
 Proceedings of FOC '78, Information Gatekeepers
"... SemiLagrangian multistep exponential integrators for index 2 differential algebraic system by ..."
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Cited by 29 (0 self)
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SemiLagrangian multistep exponential integrators for index 2 differential algebraic system by
Evaluating matrix functions for exponential integrators via Carathéodory–Fejér approximation and contour integrals
 ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
, 2007
"... Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of, where is a negative semidefinite matrix and is the exponential function or one of the related “ functions ” such as. Building on previous work by Trefethen and Gutknecht, Minchev, and Lu, w ..."
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Cited by 25 (1 self)
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Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of, where is a negative semidefinite matrix and is the exponential function or one of the related “ functions ” such as. Building on previous work by Trefethen and Gutknecht, Minchev, and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems #"$& % can be solved efficiently, e.g. by a sparse direct solver. The first method is based on best rational approximations to on the negative real axis computed via the CarathéodoryFejér procedure. Rather than using optimal poles we approximate the functions in a set of common poles, which speeds up typical computations by a factor ' of (*) + to. The second method is an application of the trapezoid rule on a Talbottype contour.
A stochastic immersed boundary method for fluidstructure dynamics at microscopic length scales
 J. Comput. Phys
, 2007
"... In modeling many biological systems, it is important to take into account the interaction of flexible structures with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work it is shown how the immersed boundary met ..."
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Cited by 24 (6 self)
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In modeling many biological systems, it is important to take into account the interaction of flexible structures with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work it is shown how the immersed boundary method of [64] for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium
THE EXPONENTIALLY CONVERGENT TRAPEZOIDAL RULE
"... Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods ..."
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Cited by 17 (3 self)
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Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
Stability analysis and applications of the exponential time differencing schemes
 J. Comput. Math
"... Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday Exponential time differencing schemes are time integration methods that can be efficiently combined with spatial spectral approximations to provide very high resolution to the smooth solutions of some linear and nonlinear part ..."
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Cited by 16 (3 self)
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Dedicated to Professor Zhongci Shi on the occasion of his 70th birthday Exponential time differencing schemes are time integration methods that can be efficiently combined with spatial spectral approximations to provide very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. We study in this paper the stability properties of some exponential time differencing schemes. We also present their application to the numerical solution of the scalar AllenCahn equation in two and three dimensional spaces. Mathematics subject classification: Key words: 1.
Computing f(A)b for matrix functions f
 In QCD and numerical analysis III
, 2005
"... Summary. For matrix functions f we investigate how to compute a matrixvector product f(A)b without explicitly computing f(A). A general method is described that applies quadrature to the matrix version of the Cauchy integral theorem. Methods specific to the logarithm, based on quadrature, and fract ..."
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Cited by 13 (4 self)
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Summary. For matrix functions f we investigate how to compute a matrixvector product f(A)b without explicitly computing f(A). A general method is described that applies quadrature to the matrix version of the Cauchy integral theorem. Methods specific to the logarithm, based on quadrature, and fractional matrix powers, based on solution of an ordinary differential equation initial value problem, are also presented 1
Solving the nonlinear Schrödinger equation using exponential integrators
"... Using the notion of integrating factors, Lawson developed a class of numerical methods for solving stiff systems of ordinary differential equations. However, the performance of these “Generalized Runge–Kutta processes” was demonstrably poorer when compared to the ETD schemes of Certaine and Nørsett, ..."
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Cited by 12 (3 self)
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Using the notion of integrating factors, Lawson developed a class of numerical methods for solving stiff systems of ordinary differential equations. However, the performance of these “Generalized Runge–Kutta processes” was demonstrably poorer when compared to the ETD schemes of Certaine and Nørsett, recently rediscovered by Cox and Matthews. The deficit is particularly pronounced when the schemes are applied to parabolic problems. In this paper we compare a fourth order Lawson scheme and a fourth order ETD scheme due to Cox and Matthews, using the nonlinear Schrödinger equation as the test problem. The primary testing parameters are degree of regularity of the potential function and the initial condition, and numerical performance is heavily dependent upon these values. The Lawson and ETD schemes exhibit significant performance differences in our tests, and we present some analysis on this.
Bseries and order conditions for exponential integrators
"... Abstract. We introduce a general format of numerical ODEsolvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of Bseries and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea ..."
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Cited by 11 (4 self)
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Abstract. We introduce a general format of numerical ODEsolvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of Bseries and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea of Zennaro [Math. Comp., 46 (1986), pp. 119–133] and define natural continuous extensions in the context of exponential integrators. This leads to a relatively easy derivation of some of the most popular recently proposed schemes. The general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces. We will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes. Finally, we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5.