Implementation of exponential Rosenbrock-type integrators

In this paper, we present a variable step size implementation of exponential Rosenbrock-type methods of orders 2, 3 and 4. These integrators require the evaluation of exponential and related functions of the Jacobian matrix. To this aim, the Real Leja Points Method is used. It is shown that the properties of this method combine well with the particular requirements of Rosenbrock-type integrators. We verify our implementation with some numerical experiments in MATLAB, where we solve semilinear parabolic PDEs in one and two space dimensions. We further present some numerical experiments in FORTRAN, where we compare our method with other methods from literature. We find a great potential of our method for non-normal matrices. Such matrices typically arise in parabolic problems with large advection in combination with moderate diffusion and mildly stiff reactions.     

The scaling and modified squaring method for matrix functions related to the exponential

In recent years, there has been a resurgence in the construction and implementation of exponential integrators, which are numerical methods specifically designed for the numerical solution of spatially discretized semi-linear partial differential equations. Exponential integrators use the matrix exponential and related matrix functions within the formulation of the numerical method. The scaling and squaring method is the most widely used method for computing the matrix exponential. The aim of this paper is to discuss the efficient and accurate evaluation of the matrix exponential and related matrix functions using a scaling and modified squaring method.     

Accurate dense output formula for exponential integrators using the scaling and squaring method

This paper proposes an accurate dense output formula for exponential integrators. The computation of matrix exponential function is a vital step in implementing exponential integrators. By scrutinizing the computational process of matrix exponentials using the scaling and squaring method, valuable intermediate results in this process are identified and then used to establish a dense output formula. Efficient computation of dense outputs by the proposed formula enables time integration methods to set their simulation step sizes more flexibly. The efficacy of the proposed formula is verified through numerical examples from the power engineering field.     

Exponential Rosenbrock integrators for option pricing

In this paper, we are concerned with the time integration of differential equations modeling option pricing. In particular, we consider the Black-Scholes equation for American options. As an alternative to existing methods, we present exponential Rosenbrock integrators. These integrators require the evaluation of the exponential and related functions of the Jacobian matrix. The resulting methods have good stability properties. They are fully explicit and do not require the numerical solution of linear systems, in contrast to standard integrators. We have implemented some numerical experiments in Matlab showing the reliability of the new method.     

Approximation of the linear combination of $\varphi$-functions using the block shift-and-invert Krylov subspace method

In this paper, we develop an algorithm in which the block shift-and-invert Krylov subspace method can be employed for approximating the linear combination of the matrix exponential and related exponential-type functions. Such evaluation plays a major role in a class of numerical methods known as exponential integrators. We derive a low-dimensional matrix exponential to approximate the objective function based on the block shift-and-invert Krylov subspace methods. We obtain the error expansion of the approximation, and show that the variants of its first term can be used as reliable a posteriori error estimates and correctors. Numerical experiments illustrate that the error estimates are efficient and the proposed algorithm is worthy of further study.     

A framework of the harmonic Arnoldi method for evaluating <Emphasis Type="Italic">φ</Emphasis>-functions with applications to exponential integrators

In recent years, a great deal of attention has been focused on exponential integrators. The important ingredient to the implementation of exponential integrators is the efficient and accurate evaluation of the so called φ-functions on a given vector. The Krylov subspace method is an important technique for this problem. For this type of method, however, restarts become essential for the sake of storage requirements or due to computational complexities of evaluating matrix function on a reduced matrix of growing size. Another problem in computing φ-functions is the lack of a clear residual notion. The contribution of this work is threefold. First, we introduce a framework of the harmonic Arnoldi method for φ-functions, which is based on the residual and the oblique projection technique. Second, we establish the relationship between the harmonic Arnoldi approximation and the Arnoldi approximation, and compare the harmonic Arnoldi method and the Arnoldi method from a theoretical point of view. Third, we apply the thick-restarting strategy to the harmonic Arnoldi method, and propose a thick-restarted harmonic Arnoldi algorithm for evaluating φ-functions. An advantage of the new algorithm is that we can compute several φ-functions simultaneously in the same search subspace after restarting. The relationship between the error and the residual of the harmonic Arnoldi approximation is also investigated. Numerical experiments show the superiority of our new algorithm over many state-of-the-art algorithms for computing φ-functions.     

On the construction of restricted-denominator exponential W-methods

In this paper we consider the practical construction of exponential W-methods for the solution of large stiff nonlinear initial value problems, based on the restricted-denominator rational approach for the computation of the functions of matrices required. This approach is employed together with the Krylov subspace method based on the Arnoldi algorithm. Two integrators are constructed and tested on some classical stiff equations arising from the semidiscretization of parabolic problems.     

Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge–Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behaviour of the new exponential integrators in comparison with some symmetric and symplectic Runge–Kutta methods in the literature.     

Rational Krylov methods in exponential integrators for European option pricing

The aim of this paper is to analyze efficient numerical methods for time integration of European option pricing models. When spatial discretization is adopted, the resulting problem consists of an ordinary differential equation that can be approximated by means of exponential Runge–Kutta integrators, where the matrix‐valued functions are computed by the so‐called shift‐and‐invert Krylov method. To our knowledge, the use of this numerical approach is innovative in the framework of option pricing, and it reveals to be very attractive and efficient to solve the problem at hand. In this respect, we propose some a posteriori estimates for the error in the shift‐and‐invert approximation of the core‐functions arising in exponential integrators. The effectiveness of these error bounds is tested on several examples of interest. They can be adopted as a convenient stopping criterion for implementing the exponential Runge–Kutta algorithm in order to perform time integration. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of higher order phase fitted variational integrators

In the present, the linear stability properties of the higher order phase fitted variational integrators are investigated. Towards this purpose, at first we calculate the eigenvalues of the amplification matrix for each method. Then, since the proposed integrators are derived specifically for the numerical integration of systems with oscillatory solutions, the linear stability analysis verifies their good behavior, when used for these problems. Finally, we test the proposed methods on several numerical examples, first with regard to their stability and secondly concerning the behavior in long term integration of highly oscillatory problems.     

Comparison of methods for evaluating functions of a matrix exponential

We compare six different categories of numerical methods for the evaluation of functions of the matrix exponential. These functions are required for exponential integrators, and are not straightforward to evaluate because they are highly susceptible to rounding errors when the matrix has small eigenvalues. The comparison takes into account both accuracy and computational time. A scaling and squaring algorithm and a diagonalisation algorithm are both found to be efficient.     

A quadrature based method for evaluating exponential-type functions for exponential methods

We present a quadrature-based method to evaluate exponential-like operators required by different kinds of exponential integrators. The method approximates these operators by means of a quadrature formula that converges like O(e ^−cK ), with K the number of quadrature nodes, and it is useful when solving parabolic equations. The approach allows also the evaluation of the associated scalar mappings. The method is based on numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm, including examples with a mass matrix and the application of the method inside the MATLAB package EXP4, an adaptive solver based on an exponential Runge–Kutta method.     

On the stability of large matrices

The distance rstab(A) of a stable matrix A to the set of unstable matrices and the norm of the exponential of matrices constitute two important topics in stability theory. We treat in this note the case of large matrices. The method proposed partitions the matrix into two blocks: a small block in which the stability is studied and a large block whose field of values is located in the complex plane. Using the information on the blocks and some results on perturbation theory, we give sufficient conditions for the stability of the original matrix, a lower bound of rstab(A) and an upper bound on the norm of the exponential of A. We illustrate these theoretical bounds on a practical test problem.     

Solving linear initial value problems by Faber polynomials

In this paper we use the theory of Faber polynomials for solving N‐dimensional linear initial value problems. In particular, we use Faber polynomials to approximate the evolution operator creating the so‐called exponential integrators. We also provide a consistence and convergence analysis. Some tests where we compare our methods with some Krylov exponential integrators are finally shown. Copyright © 2002 John Wiley & Sons, Ltd.     

Magnus integrators for solving linear-quadratic differential games

We consider Magnus integrators to solve linear-quadratic N$N$-player differential games. These problems require to solve, backward in time, non-autonomous matrix Riccati differential equations which are coupled with the linear differential equations for the dynamic state of the game, to be integrated forward in time. We analyze different Magnus integrators which can provide either analytical or numerical approximations to the equations. They can be considered as time-averaging methods and frequently are used as exponential integrators. We show that they preserve some of the most relevant qualitative properties of the solution for the matrix Riccati differential equations as well as for the remaining equations. The analytical approximations allow us to study the problem in terms of the parameters involved. Some numerical examples are also considered which show that exponential methods are, in general, superior to standard methods.     

Symmetric Exponential Integrators with an Application to the Cubic Schrödinger Equation

In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrödinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L ²-norm and/or the energy of the system.     

Geometric integration of non-autonomous linear Hamiltonian problems

Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.     

Symmetric spaces and Lie triple systems in numerical analysis of differential equations

A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. The proposed techniques has the property that all the time-steps are positive.     

Convergence of rational multistep methods of Adams-Padé type

Rational generalizations of multistep schemes, where the linear stiff part of a given problem is treated by an A-stable rational approximation, have been proposed by several authors, but a reasonable convergence analysis for stiff problems has not been provided so far. In this paper we directly relate this approach to exponential multistep methods, a subclass of the increasingly popular class of exponential integrators. This natural, but new interpretation of rational multistep methods enables us to prove a convergence result of the same quality as for the exponential version. In particular, we consider schemes of rational Adams type based on A-acceptable Padé approximations to the matrix exponential. A numerical example is also provided.