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Explicit exponential RungeKutta methods for semilinear parabolic problems
 SIAM J. Numer. Anal
"... Abstract. The aim of this paper is to analyze explicit exponential RungeKutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giv ..."
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Abstract. The aim of this paper is to analyze explicit exponential RungeKutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonsti®) order conditions for exponential RungeKutta methods, but the main interest of our paper lies in the sti ® case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we ¯nally construct methods that do not su®er from order reduction. 1. Introduction. Motivated
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 9 (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.
Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation
, 2006
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Approximation of matrix operators applied to multiple vectors
 Math. Comput. Simulation
"... In this paper we propose a numerical method for approximating the product of a matrix function with multiple vectors by Krylov subspace methods combined with a QR decomposition of these vectors. This problem arises in the implementation of exponential integrators for semilinear parabolic problems. W ..."
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Cited by 3 (0 self)
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In this paper we propose a numerical method for approximating the product of a matrix function with multiple vectors by Krylov subspace methods combined with a QR decomposition of these vectors. This problem arises in the implementation of exponential integrators for semilinear parabolic problems. We will derive reliable stopping criteria and we suggest variants using up and downdating techniques. Moreover, we show how Ritz vectors can be included in order to speed up the computation even further. By a number of numerical examples, we will illustrate that the proposed method will reduce the total number of Krylov steps significantly compared to a standard implementation if the vectors correspond to the evaluation of a smooth function at certain quadrature points. Key words: Krylov subspace methods, shift and invert Lanczos process, projection method, matrix exponential function, matrix functions, restarts, error
NORGES TEKNISKNATURVITENSKAPELIGE UNIVERSITET
"... Solving the nonlinear Schödinger equation using exponential integrators by ..."
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Solving the nonlinear Schödinger equation using exponential integrators by
†Department of Mathematical Sciences NTNU
, 2006
"... 1 Lie group methods Consider the differential equation Y ̇ = A(Y) · Y, Y (0) = Y0, (1) where Y and A(Y) are n×n matrices, A(Y) is skewsymmetric for all Y and Y0 is an orthogonal matrix. The solution of (1) is an orthogonal matrix in fact if we take the derivative w.r.t. time of Y (t)TY (t) we ob ..."
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1 Lie group methods Consider the differential equation Y ̇ = A(Y) · Y, Y (0) = Y0, (1) where Y and A(Y) are n×n matrices, A(Y) is skewsymmetric for all Y and Y0 is an orthogonal matrix. The solution of (1) is an orthogonal matrix in fact if we take the derivative w.r.t. time of Y (t)TY (t) we obtain d dt