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In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the μ-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB®/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

Exponential integrators, due to their robust stability properties, have been considered as reliable schemes for numerical solutions of stiff systems. In this paper, we propose generalized exponential time differencing (GETD) schemes for nonlinear fractional differential equations of order α ∈ (0,1). First, we improve the suboptimal performance of the multistep GETD schemes. Using graded mesh, uniform optimal convergence rates under no additional smoothness requirements are obtained. Second, we develop and analyze novel second-order and third-order accurate predictor-corrector type GETD schemes. Using linear stability analysis and numerical illustrations, we demonstrate that the newly introduced schemes have better stability properties than the multistep GETD schemes. Partial fraction decompositions of global Padé approximations for Mittag-Leffler function are used for efficient implementation. Numerical examples involving nonlinear scalar equations and stiff systems are provided to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

We investigate some theoretical and numerical aspects of a nonlinear wave equation with variable coefficient and Dirichlet and Acoustic boundary conditions. The existence and uniqueness of the solution are obtained applying the Faedo-Galerkin method with some compactness results and energy method. In addition, we prove the uniform stability of the energy. For numerical simulation, firstly we use the Crank–Nicolson Galerkin method, in which it consists of applying the finite element method in the spatial variable and the Crank–Nicolson method over time. Subsequently, in the resulting nonlinear algebraic system, for each discrete time, we apply the Newton's method without losing the convergence order. Moreover, are presented figures of the numerical solutions for the two-dimensional case, tables with error and convergence order and the numerical energy decay. These results justify the consistency between the theoretical and numerical results. [ABSTRACT FROM AUTHOR]

Abstract: A two-dimensional fluid laser–plasma integrator for stratified plasma–vacuum systems is presented. Inside a plasma, a laser pulse can be longitudinally compressed from ten or more wave-lengths to one or two cycles. However, for physically realistic simulations, transversal effects have to be included, because transversal instabilities can destroy the pulse and transversal compression in the plasma as well as focusing in vacuum allows much higher intensities to be reached. In contrast to the one-dimensional case, where a two-step implementation of the Gautschi-type exponential integrator with constant step-size turned out to be sufficient, it is essential to enable changes of the time step-size for the two-dimensional case. The use of a one-step version of the Gautschi-type integrator, being accurate of second order independent of the highest frequencies arising in the system, is proposed. In vacuum this allows to take arbitrarily large time-steps. To optimize runtime and memory requirements within the plasma, a splitting of the Laplacian is suggested. This splitting allows to evaluate the matrix functions arising in the Gautschi-type method by one-dimensional Fourier transforms. It is also demonstrated how the different variants of the scheme can be parallelized. Numerical experiments illustrate the superior performance and accuracy of the integrator compared to the standard leap-frog method. Finally, we discuss the simulation of a layered plasma–vacuum structure using the new method. [Copyright &y& Elsevier]

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation. [ABSTRACT FROM AUTHOR]

In high‐resolution numerical weather prediction models, fast‐moving acoustic waves must be treated in a stable manner. Among the implicit–explicit (IMEX) schemes used for the solution of these models, horizontally explicit, vertically implicit (HEVI) methods show good stability and scalability on massively parallel machines. In this work, we present two classes of exponential time‐differencing (ETD) methods for atmospheric models that use a HEVI splitting strategy, one being a three‐stage method with the addition of artificial diffusion, the second based on a Strang splitting approach. The stability properties of the methods are analyzed and numerical examples are provided, which compare time‐step restrictions and cost‐to‐accuracy ratios of the new methods with those for existing approaches. [ABSTRACT FROM AUTHOR]

Evaluating the action of a matrix function on a vector, that is x = f (M) v , is an ubiquitous task in applications. When M is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating x when f(z) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and M is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case M = I ⊗ A - B T ⊗ I , and v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of x. Pole selection strategies with explicit convergence bounds are given also in this case. [ABSTRACT FROM AUTHOR]

The linear complementarity system (LCS) is defined by a linear ordinary differential equation coupled with a finite-dimensional linear complementarity problem (LCP), which has many applications in engineering and economics. In this article, we reformulate the LCS with the boundary condition as an LCP in the Hilbert space of square-integrable functions, and propose a new numerical method for the LCS by using exponential Euler integrator and discontinuous Galerkin approximation. The precision of the proposed method is better than that of the existing time-stepping method in different magnitude of scale. Convergence analysis and numerical experiments are performed to support the arguments. [ABSTRACT FROM AUTHOR]

Given an $$n$$ -by- $$n$$ large sparse matrix $$A$$ with low rank or with fast decaying singular values, we propose new algorithms for consecutively approximating the functions $$\varphi_{\ell}(A)$$ as well as their 2-norm condition numbers for $$\ell=0,1,\ldots,p$$. The key is to reduce the computation of $$\varphi_{\ell}$$ -functions of the large matrix to that of $$\varphi_{\ell+1}$$ -functions of some smaller matrices of size $$r$$ -by- $$r$$ , where $$r$$ is the numerical rank of $$A$$. For storage reasons, the new algorithms need to store only two $$n$$ -by- $$r$$ sparse matrices and some $$r$$ -by- $$r$$ matrices, rather than some $$n$$ -by- $$n$$ possibly dense matrices. Some error analysis is given. Numerical experiments illustrate the numerical behavior of our new algorithms and show the practical utility of new strategies. [ABSTRACT FROM AUTHOR]

An error analysis of a splitting method applied to the Zakharov system is given. The numerical method is a Lie-Trotter splitting in time that is combined with a Fourier collocation in space to a fully discrete method. First-order convergence in time and high-order convergence in space depending on the regularity of the exact solution are shown for this method. The main challenge in the analysis is to exclude a loss of spatial regularity in the numerical solution. This is done by transforming the numerical method to new variables and by imposing a natural CFL-type restriction on the discretization parameters. [ABSTRACT FROM AUTHOR]

The generalized fractional reaction-diffusion equations which exist in the form of noninteger order partial differential equations have now found wide application for illustrating important and useful physical phenomena, such as subdiffusive and superdiffusive scenarios. The space fractional derivatives are defined in the Riesz sense on the intervals 0<α<1 and 1<α≤2. We propose robust numerical techniques, such as a spectral representation of the fractional Laplacian operator in conjunction with the exponential time differencing method, and present the equivalent relationship between the Riesz fractional derivative and fractional Laplacian operator. We apply these techniques to numerically solve a range of chaotic processes, such as the Chua’s equations, Rössler system, Lorenz and Lorenz-type systems. Simulation results revealed various complex and spatiotemporal chaos, spiral chaos, intermittent chaos and spots patterns in two-dimensional space fractional reaction-diffusion problems. [ABSTRACT FROM AUTHOR]

Krylov subspace approximations to the matrix exponential are popularly used with full orthogonalization instead of incomplete orthogonalization, even though the latter strategy is known to reduce the cost by truncating the recurrences of the modified Gram-Schmidt process. This study combines such a strategy with an adaptive step-by-step integration scheme that allows both the stepsize and the dimension of the Krylov subspace to vary. A convergence analysis is done. Numerical results on test problems drawn from systems biology and computer systems show a significant speedup over the standard implementation with full orthogonalization and fixed dimension. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge-Kutta-Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge-Kutta-Nyström methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of $$f(\mathcal{A})b$$ where $$b\in {\mathbb {R}}^{N}$$ and $$\mathcal A$$ is the sum of Kronecker products, that is $$\mathcal{A}=M_2 \otimes I + I \otimes M_1\in {\mathbb {R}}^{N\times N}$$ . Here f is a regular function such that $$f(\mathcal{A})$$ is well defined. We derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations, with special emphasis on the exponential and on completely monotonic functions, for which the new procedure becomes particularly advantageous. Our findings are illustrated by numerical experiments with typical functions used in applications. [ABSTRACT FROM AUTHOR]

We build efficient and unitary (hence stable) methods for the solution of the linear time-dependent Schrödinger equation with explicitly time-dependent potentials in a semiclassical regime. The Magnus-Zassenhaus schemes presented here are based on a combination of the Zassenhaus decomposition (Bader et al. 2014 Found. Comput. Math. 14, 689-720. (doi:10.1007/s10208-013-9182-8)) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments. [ABSTRACT FROM AUTHOR]

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric differentiation matrices for univariate finite-difference methods. We prove that, in order to sustain a skew-symmetric differentiation matrix of order $$p\ge 2$$ , a grid must satisfy $$2p-3$$ polynomial conditions. Moreover, once it satisfies these conditions, it supports a banded skew-symmetric differentiation matrix of this order and of the bandwidth $$2p-1$$ , which can be derived in a constructive manner. Some applications require not just skew-symmetry, but also that the growth in the elements of the differentiation matrix is at most linear in the number of unknowns. This is always true for our tridiagonal matrices of order 2 but need not be true otherwise, a subject which we explore further. Another subject which we examine is the existence and practical construction of grids that support skew-symmetric differentiation matrices of a given order. We resolve this issue completely for order-two methods. We conclude the paper with a list of open problems and their discussion. [ABSTRACT FROM AUTHOR]

In this paper, a high-order accurate numerical method for two-dimensional semilinear parabolic equations is presented. We apply a Galerkin-Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high-order accurate in both space and time. Optimal a priori error bound is derived in the L2-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Reduced-order models for linear time-invariant dynamical systems are considered, and the error between the full-order model and the reduced-order model solutions is characterized. Based on the analytical representation of the error, an a posteriori error indicator is proposed that combines a Krylov-based exponential integrator and an a posteriori residual-based estimate. Numerical experiments illustrate the quality of the error estimator. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Krylov subspace methods for approximating a matrix function f( A) times a vector v are analyzed in this paper. For the Arnoldi approximation to e v, two reliable a posteriori error estimates are derived from the new bounds and generalized error expansion we establish. One of them is similar to the residual norm of an approximate solution of the linear system, and the other one is determined critically by the first term of the error expansion of the Arnoldi approximation to e v due to Saad. We prove that each of the two estimates is reliable to measure the true error norm, and the second one theoretically justifies an empirical claim by Saad. In the paper, by introducing certain functions ϕ( z) defined recursively by the given function f( z) for certain nodes, we obtain the error expansion of the Krylov-like approximation for f( z) sufficiently smooth, which generalizes Saad's result on the Arnoldi approximation to e v. Similarly, it is shown that the first term of the generalized error expansion can be used as a reliable a posteriori estimate for the Krylov-like approximation to some other matrix functions times v. Numerical examples are reported to demonstrate the effectiveness of the a posteriori error estimates for the Krylov-like approximations to e v, cos( A) v and sin( A) v. [ABSTRACT FROM AUTHOR]

SUMMARY This paper is concerned with the numerical solution of symmetric large-scale Lyapunov equations with low-rank right-hand sides and coefficient matrices depending on a parameter. Specifically, we consider the situation when the parameter dependence is sufficiently smooth, and the aim is to compute solutions for many different parameter samples. On the basis of existing results for Lyapunov equations and parameter-dependent linear systems, we prove that the tensor containing all solution samples typically allows for an excellent low multilinear rank approximation. Stacking all sampled equations into one huge linear system, this fact can be exploited by combining the preconditioned CG method with low-rank truncation. Our approach is flexible enough to allow for a variety of preconditioners based, for example, on the sign function iteration or the alternating direction implicit method. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY 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. [ABSTRACT FROM AUTHOR]

SUMMARY The focus of this paper is two-dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two-scale Richards' equation-based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro-scale as a two-dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro-scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two-scale model in a completely coupled manner. Numerical simulations are presented for a two-dimensional infiltration problem. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The theme of this paper is the construction of finite difference approximations to the first derivative in the presence of Dirichlet boundary conditions. Stable implementation of splitting-based discretization methods for the convection–diffusion equation requires the underlying matrix to be skew symmetric and this turns out to be a surprisingly restrictive condition. We prove that no skew-symmetric approximation on an equidistant grid may exceed order 2. Once a nonequidistant grid is allowed, this barrier can be breached. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the error analysis for the two-step extended Runge-Kutta-Nyström-type (TSERKN) methods [Comput. Phys. Comm. 182 (2011) 2486-2507] for multi-frequency and multidimensional oscillatory systems y″( t) + My( t) = f( t, y( t)), where high-frequency oscillations in the solutions are generated by the linear part My( t). TSERKN methods extend the two-step hybrid methods [IMA J. Numer. Anal. 23 (2003) 197-220] by reforming both the internal stages and the updates so that they are adapted to the oscillatory properties of the exact solutions. However, the global error analysis for the TSERKN methods has not been investigated. In this paper we construct a new three-stage explicit TSERKN method of order four and present the global error bound for the new method, which is proved to be independent of ∥ M∥ under suitable assumptions. This property of our new method is very important for solving highly oscillatory systems (1), where ∥ M∥ may be arbitrarily large. We also analyze the stability and phase properties for the new method. Numerical experiments are included and the numerical results show that the new method is very competitive and promising compared with the well-known high quality methods proposed in the scientific literature. [ABSTRACT FROM AUTHOR]

The numerical integration of Hamiltonian systems with multi-frequency and multidimensional oscillatory solutions is encountered frequently in many fields of the applied sciences. In this paper, we firstly summarize the extended Runge-Kutta-Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1873-1887, (2010)) for multi-frequency and multidimensional oscillatory systems and restate the order conditions and symplecticity conditions for the explicit ERKN methods. Secondly, we devote to exploring the explicit symplectic multi-frequency and multidimensional ERKN methods of order five based on the symplecticity conditions and order conditions. A five-stage explicit symplectic multi-frequency and multidimensional ERKN method of order five with some small residuals is proposed and its stability and phase properties are analyzed. It is shown that the new method is dispersive of order six. Numerical experiments are carried out and the numerical results demonstrate that the new method is much more efficient than the methods appeared in the scientific literature. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, the numerical evaluation of matrix functions expressed in partial fraction form is addressed. The shift-and-invert Krylov method is analyzed, with special attention to error estimates. Such estimates give insights into the selection of the shift parameter and lead to a simple and effective restart procedure. Applications to the class of Mittag-Leffler functions are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY We propose a time-exact Krylov-subspace-based method for solving linear ordinary differential equation systems of the form y ′ = − Ay + g( t) and y ′ ′ = − Ay + g( t), where y( t) is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term g( t), constructed with the help of the truncated singular value decomposition. The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Because both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We follow up on our previous work (Le Bris, C. & Legoll, F. (2007) Derivation of symplectic numerical schemes for highly oscillatory Hamiltonian systems. C. R. Acad. Sci., 344, 277–282 and Le Bris, C. & Legoll, F. (2010) Integrators for highly oscillatory Hamiltonian systems: an homogenization approach. Discrete Continuous Dyn. Syst. Ser. B, 13, 347–373) which presented a possible approach for deriving symplectic schemes for a certain class of highly oscillatory Hamiltonian systems. The approach considers the Hamilton–Jacobi form of the equations of motion, formally homogenizes it and infers an appropriate symplectic integrator for the original system. In Le Bris & Legoll (2007) and Le Bris & Legoll (2010), the case of a system exhibiting a single constant fast frequency was considered. The present work successfully extends the approach to systems that have either one varying fast frequency or several constant fast frequencies. Some related issues are also examined. [ABSTRACT FROM AUTHOR]

We first review the known mathematical results concerning the Kadomtsev-Petviashvili type equations. Then we perform numerical simulations to analyze various qualitative properties of the equations: blow-up versus long time behavior, stability and instability of solitary waves. [ABSTRACT FROM AUTHOR]

We consider the approximation of operator functions in resolvent Krylov subspaces. Besides many other applications, such approximations are currently of high interest for the approximation of φ-functions that arise in the numerical solution of evolution equations by exponential integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behaviour if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we analyse a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators whose field of values lies in the left half plane. In contrast to standard Krylov methods, the convergence will be independent of the norm of the discretised operator and thus of the spatial discretisation. We will discuss efficient implementations for finite element discretisations and illustrate our analysis with numerical experiments. [ABSTRACT FROM AUTHOR]

The Liouville-Green (WKB) asymptotic theory is used along with the Borůvka's transformation theory, to obtain asymptotic approximations of 'phase functions' for second-order linear differential equations, whose coefficients are asymptotically polynomial. An efficient numerical method to compute zeros of solutions or even the solutions themselves in such highly oscillatory problems is then derived. Numerical examples, where symbolic manipulations are also used, are provided to illustrate the performance of the method. [ABSTRACT FROM AUTHOR]

SUMMARY A fast exponential time integration scheme is considered for pricing European and double barrier options in jump-diffusion models. After spatial discretization, the option pricing problem is transformed into the product of a matrix exponential and a vector, while the matrix bears a Toeplitz structure. The shift-and-invert Arnoldi method is then employed for fast approximation to such operation. Owing to the Toeplitz structure, the computational cost can be reduced by the fast Fourier transform. Furthermore, the discretized form of option pricing problem satisfies a given condition such that the error bound of the shift-and-invert Arnoldi approximation is unrelated to the norm of the matrix. Numerical tests are carried out to compare the proposed method with other option pricing methods. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Lanczos method with shift-invert technique is exploited to approximate the symmetric positive semidefinite Toeplitz matrix exponential. The complexity is lowered by the Gohberg-Semencul formula and the fast Fourier transform. Application to the numerical solution of an integral equation is studied. Numerical experiments are carried out to demonstrate the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

General stationary iterative methods with a singular matrix M for solving range-Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega–Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range-Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Questions concerning the accuracy of numerical methods for differential equations are often analysed using B-series and other formulations based on rooted trees. The analysis of numerical methods, such as Rosenbrock and certain exponential methods, requires an additional algebraic structure to represent the direct use of Jacobian matrices in the computation. It is shown how this can be done within a context containing a broad review of the existing B-series formulation. [ABSTRACT FROM PUBLISHER]

We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estimates are derived. Numerical experiments are presented. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator φ( hA) v, φ ( z) = ( e z − 1)/ z. In this way, it is possible to approximate φ ( hA) v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to “higher” order φ k functions, k≥0. [ABSTRACT FROM AUTHOR]

We investigate the numerical solution of large-scale Lyapunov equations with the sign function method. Replacing the usual matrix inversion, addition, and multiplication by formatted arithmetic for hierarchical matrices, we obtain an implementation that has linear-polylogarithmic complexity and memory requirements. The method is well suited for Lyapunov operators arising from FEM and BEM approximations to elliptic differential operators. With the sign function method it is possible to obtain a low-rank approximation to a full-rank factor of the solution directly. The task of computing such a factored solution arises, e.g., in model reduction based on balanced truncation. The basis of our method is a partitioned Newton iteration for computing the sign function of a suitable matrix, where one part of the iteration uses formatted arithmetic while the other part directly yields approximations to the full-rank factor of the solution. We discuss some variations of our method and its application to generalized Lyapunov equations. Numerical experiments show that the method can be applied to problems of order up to [InlineMediaObject not available: see fulltext.](105) on workstations. [ABSTRACT FROM AUTHOR]

The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kernel polynomials are considered. Theoretical results as well as numerical experiments illustrate the effectiveness of our approach. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The Yee-method is a simple and elegant way of solving the time-dependent Maxwell's equations. On the other hand, this method has some inherent drawbacks too. The main one is that its stability requires a very strict upper bound for the possible time-steps. This is why, during the last decade, the main goal was to construct such methods that are unconditionally stable. This means that the time-step can be chosen based only on accuracy instead of stability considerations. In this paper we give a uniform treatment of methods that use the same spatial staggered grid approximation as the classical Yee-method. Three other numerical methods are discussed: the Namiki–Zheng–Chen–Zhang alternating direction implicit method (NZCZ), the Kole–Figge-de Raedt method (KFR) and a Krylov-space method. All methods are discussed with non-homogeneous material parameters. We show how the existing finite difference numerical methods are based on the approximation of a matrix exponential. With this formulation we prove the unconditional stability of the NZCZ method without any computer algebraic tool. Moreover, we accelerate the Krylov-space method with a skew-symmetric formulation of the semi-discretized equations. Our main goal is to compare the methods from the point of view of the computational speed. This question is investigated in ID numerical tests. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Extended one-step schemes of exponential type are introduced for the numerical solution of stiff initial-value problems. These schemes are uniformly convergent of third and fourth orders of accuracy. In addition, we show that these schemes are optimal when ℇ → 0. Numerical results and comparisons with other schemes are presented. [ABSTRACT FROM AUTHOR]

This paper deals with two-step W-methods for stiff initial value problems which allow the parallel solution of the stage equations. Special methods of order 3 and 4 are tested for both sequential and parallel implementations and compared with efficient sequential codes. The results show that the methods are already competitive in a sequential implementation and have a high level of parallelization especially in combination with Krylov solvers for linear equations of large dimensions. [ABSTRACT FROM AUTHOR]

In this paper we compare Krylov subspace methods with Chebyshev series expansion for approximating the matrix exponential operator on large, sparse, symmetric matrices. Experimental results upon negative-definite matrices with very large size, arising from (2D and 3D) FE and FD spatial discretization of linear parabolic PDEs, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques, especially when memory bounds do not allow the storage of all Ritz vectors. We also discuss the sensitivity of Chebyshev convergence to extreme eigenvalue approximation, as well as the reliability of various a priori and a posteriori error estimates for both methods. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss iterative and noniterative splitting methods, in theory and application, to solve stochastic Burgers' equations in an inviscid form. We present the noniterative splitting methods, which are given as Lie–Trotter and Strang-splitting methods, and we then extend them to deterministic–stochastic splitting approaches. We also discuss the iterative splitting methods, which are based on Picard's iterative schemes in deterministic–stochastic versions. The numerical approaches are discussed with respect to decomping deterministic and stochastic behaviours, and we describe the underlying numerical analysis. We present numerical experiments based on the nonlinearity of Burgers' equation, and we show the benefits of the iterative splitting approaches as efficient and accurate solver methods. [ABSTRACT FROM AUTHOR]