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54
Exponential integrators
, 2010
"... In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential eq ..."
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Cited by 68 (5 self)
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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in
Longtime energy conservation of numerical methods for oscillatory differential equations
 SIAM J. Numer. Anal
"... Longtime energy conservation of numerical methods for oscillatory differential equations HAIRER, Ernst, LUBICH, Christian We consider secondorder differential systems where highfrequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss ..."
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Cited by 47 (5 self)
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Longtime energy conservation of numerical methods for oscillatory differential equations HAIRER, Ernst, LUBICH, Christian We consider secondorder differential systems where highfrequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss two invariants of the system that determines its coefficients. These invariants are related to the total energy and the oscillatory harmonic energy of the original system. For the numerical solution we study a class of symmetric methods %of order 2 that discretize the linear part without error. We are interested in the case where the product of the step size with the highest frequency can be large. In the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system. This allows us to prove the nearconservation of the total and the oscillatory energy over very long time intervals.
A Reversible Averaging Integrator for Multiple TimeScale Dynamics
, 2001
"... This paper describes a new reversible staggered timestepping method for simulating longterm dynamics formulated on two or more timescales. By assuming a partitioning into fast and slow variables, it is possible to design an integrator that (1) averages the force acting on the slow variables over th ..."
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Cited by 17 (2 self)
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This paper describes a new reversible staggered timestepping method for simulating longterm dynamics formulated on two or more timescales. By assuming a partitioning into fast and slow variables, it is possible to design an integrator that (1) averages the force acting on the slow variables over the fast motions and (2) resolves the fast variables on a finer timescale than the others. The method is described for Hamiltonian systems, but could be adapted to certain types of nonHamiltonian reversible systems. Key Words: Timereversible multiple timescale integrator, Verlet method, multirate methods
Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review
"... Summary. Numerical methods for oscillatory, multiscale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time or statedependent frequencies is emphasized. Trig ..."
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Cited by 13 (1 self)
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Summary. Numerical methods for oscillatory, multiscale Hamiltonian systems are reviewed. The construction principles are described, and the algorithmic and analytical distinction between problems with nearly constant high frequencies and with time or statedependent frequencies is emphasized. Trigonometric integrators for the first case and adiabatic integrators for the second case are discussed in more detail. 1
Numerical energy conservation for multifrequency oscillatory differential equations
 BIT
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Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator
"... We propose an exponential wave integrator sine pseudospectral (EWISP) method for the nonlinear Schrödinger equation (NLS) with wave operator (NLSW), and carry out rigorous error analysis. The NLSW is NLS perturbed by the wave operator with strength described by a dimensionless parameter ε ∈ (0,1] ..."
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Cited by 11 (8 self)
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We propose an exponential wave integrator sine pseudospectral (EWISP) method for the nonlinear Schrödinger equation (NLS) with wave operator (NLSW), and carry out rigorous error analysis. The NLSW is NLS perturbed by the wave operator with strength described by a dimensionless parameter ε ∈ (0,1]. As ε → 0 +, the NLSW converges to the NLS and for the small perturbation, i.e. 0 < ε ≪ 1, the solution of the NLSW differs from that of the NLS with a function oscillating in time with O(ε 2)wavelength at O(ε 2) and O(ε 4) amplitudes for illprepared and wellprepared initial data, respectively. This rapid oscillation in time brings significant difficulties in designing and analyzing numerical methods with error bounds uniformly in ε. In this work, we show that the proposed EWISP possesses the optimal uniform error bounds at O(τ 2) and O(τ) in τ (time step) for wellprepared initial data and illprepared initial data, respectively, and spectral accuracy in h (mesh size) for the both cases, in the L 2 and semiH 1 norms. This result significantly improves the error bounds of the finite difference methods for the NLSW. Our approach involves a careful study of the error propagation, cutoff of the nonlinearity and the energy method. Numerical examples are provided to confirm our theoretical analysis.
Adiabatic integrators for highly oscillatory secondorder linear differential equations with timevarying eigendecomposition, BIT 45
, 2005
"... Numerical integrators for secondorder differential equations with timedependent high frequencies are proposed and analysed. We derive two such methods, called the adiabatic midpoint rule and the adiabatic Magnus method. The integrators are based on a transformation of the problem to adiabatic vari ..."
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Cited by 6 (1 self)
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Numerical integrators for secondorder differential equations with timedependent high frequencies are proposed and analysed. We derive two such methods, called the adiabatic midpoint rule and the adiabatic Magnus method. The integrators are based on a transformation of the problem to adiabatic variables and an expansion technique for the oscillatory integrals. They can be used with far larger step sizes than those required by traditional schemes, as is illustrated by numerical experiments. We prove secondorder error bounds with step sizes significantly larger than the almostperiod of the fastest oscillations. AMS subject classification (2000): 65L05, 65L70. Key words: oscillatory problem, numerical integrator, longtimestep method, multiple
Pantograph and Catenary Dynamics: A Benchmark Problem and Its Numerical Solution
, 1998
"... Coupled systems of partial differential equations (PDE's) and differentialalgebraic equations (DAE's) are of actual interest in various practical applications. From this point of view we have recently studied the interaction of pantograph and catenary in high speed trains [15]. To stimula ..."
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Cited by 6 (0 self)
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Coupled systems of partial differential equations (PDE's) and differentialalgebraic equations (DAE's) are of actual interest in various practical applications. From this point of view we have recently studied the interaction of pantograph and catenary in high speed trains [15]. To stimulate further research on this topic we formulate in the present paper a simplified model problem that reflects basic parts of the nonlinear dynamics in the technical system pantograph/catenary. Following the method of lines the equations of motion are semidiscretized in space using finite differences. For time discretization, typical DAE techniques are applied such as index reduction, projection steps and handling of systems with varying structure. 1 Introduction The increasing speed of modern high speed trains causes challenging problems in the design of trains, tracks, and catenaries. One of the most sensitive parts is the transmission of electrical energy via catenary and pantograph, see Fig. 1, si...
Exponential Runge–Kutta methods for the Schrödinger equation
 Appl. Numer. Math
"... Abstract We consider exponential RungeKutta methods of collocation type, and use them to solve linear and semilinear Schrödinger Cauchy problems on the ddimensional torus. We prove that in both cases (linear and nonlinear) and with suitable assumptions, sstage methods are of order s and we give ..."
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Cited by 6 (0 self)
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Abstract We consider exponential RungeKutta methods of collocation type, and use them to solve linear and semilinear Schrödinger Cauchy problems on the ddimensional torus. We prove that in both cases (linear and nonlinear) and with suitable assumptions, sstage methods are of order s and we give sufficient conditions to achieve orders s + 1 and s + 2. We show and explain the effects of resonant time steps that occur when solving linear Schrödinger problems on a finite time interval with such methods. This work is inspired by