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In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving systems of equations where the forcing term is comprised of several additive nonlinear terms. We analyze the stability, convergence, and efficiency of the new integrators and compare their performance with existing schemes for such systems using several numerical examples. We also propose a novel approach to visualizing the linear stability of the partitioned schemes, which provides a more intuitive way to understand and compare the stability properties of various schemes. Our new integrators are A-stable, second-order methods that require only one call to the linear system solver and one exponential-like matrix function evaluation per time step. [ABSTRACT FROM AUTHOR]

In this paper, we study a new splitting method for the semi-linear system of ordinary differential equation, where the linear part is stiff. Firstly, the stiff part is split into two parts. The first stiff part, that is called the stiff-cutter and expected to be easily inverted, is implicitly treated. The second stiff part and the remaining nonlinear part are explicitly treated. Therefore, such stiff-cut method can be fast implemented and save the CPU time. Theoretically, we rigorously prove that the proposed method is unconditionally stable and convergent, if the stiff-cutter is chosen to be well-matched in the stiff part. As an application, we apply our method to solve a spatial-fractional reaction-diffusion equation and give a way for how to choose a suitable stiff-cutter. Finally, numerical experiments are carried out to illustrate the accuracy and efficiency of the proposed stiff-cut method. [ABSTRACT FROM AUTHOR]

We develop two new sets of stable, rank-adaptive dynamically orthogonal Runge--Kutta (DORK) schemes that capture the high-order curvature of the nonlinear low-rank manifold. The DORK schemes asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while preserving mode continuity using newly derived retractions. We show that arbitrarily high-order optimal perturbative retractions can be obtained, and we prove that these new retractions are stable. In addition, we demonstrate that repeatedly applying retractions yields a gradient-descent algorithm on the low-rank manifold that converges superlinearly when approximating a low-rank matrix. When approximating a higher-rank matrix, iterations converge linearly to the best low-rank approximation. We then develop a rank-adaptive retraction that is robust to overapproximation. Building off of these retractions, we derive two rank-adaptive integration schemes that dynamically update the subspace upon which the system dynamics are projected within each time step: the stable, optimal dynamically orthogonal Runge--Kutta (so-DORK) and gradient-descent dynamically orthogonal Runge--Kutta (gd-DORK) schemes. These integration schemes are numerically evaluated and compared on an ill-conditioned matrix differential equation, an advection-diffusion partial differential equation, and a nonlinear, stochastic reaction-diffusion partial differential equation. Results show a reduced error accumulation rate with the new stable, optimal and gradient-descent integrators. In addition, we find that rank adaptation allows for highly accurate solutions while preserving computational efficiency. [ABSTRACT FROM AUTHOR]

Integral representations of solutions of linear multiplicatively perturbed differential equations are obtained, the diffusion part of which is bilinear on the state vector and the vector of independent Wiener processes. Equations of such class serve as models of stochastic systems with control functioning under conditions of parametric uncertainty or undesirable influence of external disturbances. The concepts and analytical apparatus of the theory of Lie algebras are used to find integral representations and fundamental matrices of the equations. [ABSTRACT FROM AUTHOR]

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]

This paper is devoted to the construction and analysis of uniformly accurate (UA) nested Picard iterative integrators (NPI) for highly oscillatory second-order differential equations. The equations involve a dimensionless parameter ε ∈ (0,1], and their solutions are highly oscillatory in time with wavelength at O (ε 2) , which brings severe burdens in numerical computation when ε ≪ 1. In this work, we first propose two NPI schemes for solving a differential equation. The schemes are uniformly first- and second-order accurate for all ε ∈ (0,1]. Moreover, they are super convergent when the time-step size is smaller than ε2. Then, the schemes are generalized to a system of differential equations with the same uniform accuracies. Error bounds are rigorously established and numerical results are reported to confirm the error estimates. [ABSTRACT FROM AUTHOR]

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. Therefore, exponential Euler differences for fractional-order differential models are novel and significant. In allusion to fuzzy Cohen–Grossberg neural networks of fractional order, this paper establishes a Mittag–Leffler Euler difference model on the basis of the constant variation formula in fractional calculus. In the second place, the existence of a unique global bounded solution and equilibrium point is studied by using the contractive mapping principle. Thirdly, global exponential stability and synchronisation of the derived difference models are investigated by employing some inequality techniques and reductio. At last, some illustrative examples and numerical simulations are employed to demonstrate the main results of this paper. [ABSTRACT FROM AUTHOR]

Exponential Euler differences for semi-linear differential equations of first order have got rapid development in the past few years and a variety of exponential Euler difference methods have become very significant researching topics. In allusion to fuzzy genetic regulatory networks of fractional order, this paper firstly establishes a novel difference method called Mittag-Leffler Euler difference, which includes the exponential Euler difference. In the second place, the existence of a unique global bounded solution and equilibrium point, global exponential stability and synchronization of the derived difference models are investigated. Compared with the classical fractional Euler differences, fuzzy Mittag-Leffler discrete-time genetic regulatory networks can better depict and retain the dynamic characteristics of the corresponding continuous-time models. What's more important is that it starts a new avenue for studying discrete-time fractional-order systems and a set of theories and methods is constructed in studying Mittag-Leffler discrete models. [ABSTRACT FROM AUTHOR]

Nonlinear differential equations (DEs) are used in a wide range of scientific problems to model complex dynamic systems. The differential equations often contain unknown parameters that are of scientific interest, which have to be estimated from noisy measurements of the dynamic system. Generally, there is no closed-form solution for nonlinear DEs, and the likelihood surface for the parameter of interest is multi-modal and very sensitive to different parameter values. We propose a Bayesian framework for nonlinear DE systems. A flexible nonparametric function is used to represent the dynamic process such that expensive numerical solvers can be avoided. A sequential Monte Carlo algorithm in the annealing framework is proposed to conduct Bayesian inference for parameters in DEs. In our numerical experiments, we use examples of ordinary differential equations and delay differential equations to demonstrate the effectiveness of the proposed algorithm. We developed an R package that is available at . Supplementary files for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper, we consider highly oscillatory second-order differential equations x ¨ (t) + Ω 2 x (t) = g (x (t)) with a single frequency confined to the linear part, and Ω is singular. It is known that the asymptotic-numerical solvers are an effective approach to numerically solve the highly oscillatory problems. Unfortunately, however, the existing asymptotic-numerical solvers fail to apply to the highly oscillatory second-order differential equations when Ω is singular. We propose an efficient improvement on the existing asymptotic-numerical solvers, so that the asymptotic-numerical solvers can be able to solve this class of highly oscillatory ordinary differential equations. The error estimation of the asymptotic-numerical solver is analyzed and nearly conservation of the energy in the Hamiltonian case is proved. Two numerical examples including the Fermi–Pasta–Ulam problem are implemented to show the efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

This paper is devoted to the explicit pseudo two-step exponential Runge–Kutta (EPTSERK) methods for the numerical integration of first-order ordinary differential equations. These methods inherit the structure of explicit pseudo two-step Runge–Kutta methods and explicit exponential Runge–Kutta methods. We analyze the order conditions and the global errors of the new methods. The new methods are of order s + 1 with s-stages for some suitable nodes. Numerical experiments are reported to show the convergence and the efficiency of the new methods. [ABSTRACT FROM AUTHOR]

This paper introduces the Runge–Kutta Chebyshev descent method (RKCD) for strongly convex optimisation problems. This new algorithm is based on explicit stabilised integrators for stiff differential equations, a powerful class of numerical schemes that avoid the severe step size restriction faced by standard explicit integrators. For optimising quadratic and strongly convex functions, this paper proves that RKCD nearly achieves the optimal convergence rate of the conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the condition number of the quadratic function worsens. It is established that this optimal rate is obtained also for a partitioned variant of RKCD applied to perturbations of quadratic functions. In addition, numerical experiments on general strongly convex problems show that RKCD outperforms Nesterov's accelerated gradient descent. [ABSTRACT FROM AUTHOR]

In the last few decades, Runge-Kutta-Nyström (RKN) methods have made significant progress and the study of RKN-type methods for solving highly oscillatory differential equations has received a great deal of attention. In this paper, from the point of view of geometric integration, the oscillation-preserving behaviour of the existing RKN-type methods in the literature are analysed in detail. To this end, it is convenient to introduce the concept of oscillation preservation for RKN-type methods. It turns out that if both the internal stages and the updates of an RKN-type method respect the qualitative and global features of the highly oscillatory solution, then the method is oscillation preserving. Since the internal stages of standard RKN and adapted RKN (ARKN) methods are inimical to the oscillation-preserving structure, neither ARKN methods nor the symplectic and symmetric RKN methods, and standard RKN methods are oscillation preserving. Other concerns relating to oscillation preservation are also considered. In particular, we are concerned with the computational issues for efficiently solving semi-discrete wave equations such as semi-discrete Klein-Gordon (KG) equations and damped sine-Gordon equations. The results of numerical experiments show the importance of the oscillation-preserving property for an RKN-type method and the remarkable superiority of oscillation-preserving integrators when applied to nonlinear multi-frequency highly oscillatory systems. [ABSTRACT FROM AUTHOR]

This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell's equations and finite elements with mass lumping for the acoustic wave equation. [ABSTRACT FROM AUTHOR]

It is well known the order reduction phenomenon which arises when exponential methods are used to integrate time-dependent initial boundary value problems, so that the classical order of these methods is reduced. In particular, this subject has been recently studied for Lie–Trotter and Strang exponential splitting methods, and the order observed in practice has been exactly calculated. In this article, a technique is suggested to avoid that order reduction. We deal directly with nonhomogeneous time-dependent boundary conditions, without having to reduce the problem to the homogeneous ones. We give a thorough error analysis of the full discretization and justify why the computational cost of the technique is negligible in comparison with the rest of the calculations of the method. Some numerical results for dimension splittings are shown, which corroborate that much more accuracy is achieved. [ABSTRACT FROM AUTHOR]

An error analysis of symmetric trigonometric integrators applied to highly oscillatory linear second-order differential equations is given. Second-order convergence is shown uniformly in the high frequencies under a finite-energy condition on the exact solution. The main novelty is the concept of the proof itself, which is based on the interpretation of trigonometric integrators as splitting methods for averaged differential equations. This allows one to combine techniques for splitting methods with those for trigonometric integrators. For the bound of the global error, cancellations in the error accumulation have to be studied carefully. [ABSTRACT FROM AUTHOR]

Exponential Lawson methods are well known to have a severe order reduction when integrating stiff problems. In a previous article, the precise order observed with Lawson methods when integrating linear problems is justified in terms of different conditions of annihilation on the boundary. In fact, the analysis of convergence with all exponential methods when applied to parabolic problems has always been performed under assumptions of vanishing boundary conditions for the solution. In this article, we offer a generalization of Lawson methods to approximate problems with nonvanishing and even time-dependent boundary values. This technique is cheap and allows to avoid completely order reduction independently of having vanishing or nonvanishing boundary conditions. [ABSTRACT FROM AUTHOR]

We apply the modulated Fourier expansion to a class of second order differential equations which consists of an oscillatory linear part and a nonoscillatory nonlinear part, with the total energy of the system possibly unbounded when the oscillation frequency grows. We comment on the difference between this model problem and the classical energy bounded oscillatory equations. Based on the expansion, we propose the multiscale time integrators to solve the ODEs under two cases: the nonlinearity is a polynomial or the frequencies in the linear part are integer multiples of a single generic frequency. The proposed schemes are explicit and efficient. The schemes have been shown from both theoretical and numerical sides to converge with a uniform second order rate for all frequencies. Comparisons with popular exponential integrators in the literature are done. [ABSTRACT FROM AUTHOR]

The chemical master equation is a differential equation to model stochastic reaction systems. Its solutions are nonnegative and $$\ell ^1$$ -contractive which is inherently related to their interpretation as probability densities. In this note, numerical discretizations of arbitrarily high order are discussed and analyzed that preserve both of these properties simultaneously and without any restriction on the discretization parameters. [ABSTRACT FROM AUTHOR]

A Lie-group differential algebraic equations (LGDAE) method, which is developed for solving differential-algebraic equations, is a simple and effective algorithm based on the Lie group GL(n,R) and the Newton iterative scheme. This paper deepens the theoretical foundation of the LGDAE method and widens its practical applications to solve nonlinear mechanical systems with nonholonomic constraints. After obtaining the closed-form formulation of elements of a one-parameter group GL(n,R) and refining the algorithm of the LGDAE method, this differential-algebraic split method is applied to solve nine problems of nonholonomic mechanics in order to evaluate its accuracy and efficiency. Numerical computations of the LGDAE method exhibit the preservation of the nonholonomic constraints with an error smaller than 10−10. Comparing the closed-form solutions demonstrates that the numerical results obtained are highly accurate, indicating that the present scheme is promising. [ABSTRACT FROM AUTHOR]

This paper studies explicit multi-frequency symmetric extended Runge-Kutta-Nyström (ERKN) integrators tailored to numerically computing the multi-frequency and multidimensional oscillatory reversible second-order differential equations $$q''(t)+Mq(t)=f\big (q(t)\big )$$ . We establish the symmetry conditions in a simplified way for multi-frequency ERKN integrators. Five explicit multi-frequency symmetric ERKN integrators are derived based on the simplified symmetry conditions. The arbitrary high-order explicit multi-frequency symmetric ERKN integrators can be achieved by the application of the symmetric composition. The stability and phase properties of the new integrators are discussed. Five numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new explicit multi-frequency symmetric integrators when applied to the multi-frequency and multidimensional oscillatory reversible second-order differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with multidimensional exponential fitting modified Runge-Kutta-Nyström (MEFMRKN) methods for the system of oscillatory second-order differential equations q″( t)+ Mq( t)= f( q( t)), where M is a d× d symmetric and positive semi-definite matrix and f( q) is the negative gradient of a potential scalar U( q). We formulate MEFMRKN methods and show clearly the relationship between MEFMRKN methods and multidimensional extended Runge-Kutta-Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1955-1962, ). Taking into account the fact that the oscillatory system is a separable Hamiltonian system with Hamiltonian $H(p,q)=\frac{1}{2}p^{T}p+ \frac{1}{2}q^{T}Mq+U(q)$, we derive the symplecticity conditions for the MEFMRKN methods. Two explicit symplectic MEFMRKN methods are proposed. Numerical experiments accompanied demonstrate that our explicit symplectic MEFMRKN methods are more efficient than some well-known numerical methods appeared in the scientific literature. [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]

We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art. [ABSTRACT FROM AUTHOR]

Recent studies of diffusion processes in highly heterogeneous, fractal-like media highlight that the traditional diffusion equation may not adequately describe the movement of water in the pore network because of the evidently anomalous transport phenomena. It is postulated here that the diffusion of moisture in the cell walls of wood is anomalous (non-Fickian), and this motivates the use of a moisture potential defined in terms of a fractional-in-space operator involving the Laplacian raised to a fractional index. A coupled anomalous transport model for temperature and moisture content is then derived to simulate the absorption of water in the cell walls of wood. This model is discretized in space using the finite volume method to produce a large system of ordinary differential equations, which is advanced in time using a second-order exponential Euler method. A novelty of our computational strategy is the way in which the flux boundary condition involving the moisture potential is implemented to ensure an accurate global mass balance is achieved at each time step. The model is validated against experimental data available for beech, where good agreement is observed. [ABSTRACT FROM AUTHOR]

We give a geometric framework for analysing iterative methods on singular linear systems A= and apply them to Krylov subspace methods. The idea is to decompose the method into the ℛ( A) component and its orthogonal complement ℛ( A), where ℛ( A) is the range of A. We apply the framework to GMRES, GMRES( k) and GCR( k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two-point boundary value problems of an ordinary differential equation. Copyright © 2010 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]

Based on the Magnus integrator method established in linear dynamic systems, an efficiently improved modified Magnus integrator method was proposed for the second-order dynamic systems with time-dependent high frequencies. Firstly, the second-order dynamic system was reformulated as the first-order system and the frame of reference was transfered by introducing new variables so that highly oscillatory behaviour inherits from the entries in the meantime. Then the modified Magnus integrator method based on local linearization was appropriately designed for solving the above new form and some improved also were presented. Finally, numerical examples show that the proposed methods appear to be quite adequate for integration for highly oscillatory dynamic systems including Hamiltonian systems problem with long time and effectiveness [ABSTRACT FROM AUTHOR]

This paper develops weak exponential schemes for the numerical solution of stochastic differential equations (SDEs) with additive noise. In particular, this work provides first and second-order methods which use at each iteration the product of the exponential of the Jacobian of the drift term with a vector. The article also addresses the rate of convergence of the new schemes. Moreover, numerical experiments illustrate that the numerical methods introduced here are a good alternative to the standard integrators for the long time integration of SDEs whose solutions by the common explicit schemes exhibit instabilities. [ABSTRACT FROM PUBLISHER]

Summary. This paper studies a numerical method for second-order oscillatory differential equations in which high-frequency oscillations are generated by a linear time- and/or solution-dependent part. For constant linear part, it is known that the method allows second-order error bounds independent of the product of the step-size with the frequencies and is therefore a long-time-step method. Most real-world problems are not of that kind and it is important to study more general equations. The analysis in this paper shows that one obtains second-order error bounds even in the case of a time- and/or solution-dependent linear part if the matrix is evaluated at averaged positions. [ABSTRACT FROM AUTHOR]

In this article, we introduce a low cost method for integrating large dimensional linear initial value problems based on the Arnoldi algorithm for matrix functions. We also describe a very efficient step-size control technique and compare a resulting algorithm of order 4 with the standard Matlab solvers on linear stiff problems arising from the discretization of parabolic equations. [ABSTRACT FROM AUTHOR]

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation. [ABSTRACT FROM AUTHOR]

In this paper, the local linearization method for the approximate computation of the prediction and filtering estimates of continuous-discrete state space models is extended to the general case of non-linear non-autonomous models with multiplicative noise. The approximate prediction and filter estimates are obtained by applying the optimal linear filter to the piecewise linear state space model that emerges from a local linearization of both the non-linear state equation and the non-linear measurement equation. In addition, the solutions of the differential equations that describe the evolution of the first two conditional moments between observations are obtained, and an algorithm for their numerical computation is also given. The performance of the LL filters is illustrated by mean of numerical experiments. [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]

[p]Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater interest recently, within the context of geometric integration[/em] and discretization methods on manifolds based on the use of Lie-group actions.[/p][p]In the present paper we study the approximation of the matrix exponential in a particular context: given a Lie group G and its Lie algebra g , we seek approximants F[/em](t B[/em]) of exp(t B[/em]) such that F[/em](t B[/em]) ∈ G if B[/em] ∈ g. Having fixed a basis V[/em][inf]1[/inf], ... , V[/em][inf]d[/inf] of g, we write F[/em](t B[/em]) as a composition of exponentials of the type exp(α[/em][inf]i[/inf] (t[/em]) V[/em][inf]i[/inf]), where α[/em][inf]i[/inf] for i[/em] = 1, 2, ... , d[/em] are scalar functions. In this manner it becomes possible to increase the order of the approximation without increasing the number of exponentials to evaluate and multiply together. We study order conditions and implementation details and conclude the paper with some numerical experiments.[/p] [ABSTRACT FROM AUTHOR]

Multistep methods for the numerical solution of ordinary differential equations are an important class of applied mathematical techniques. This paper is motivated by recently reported advances in semi-implicit numerical integration methods, multistep and extrapolation solvers. Here we propose a novel type of multistep extrapolation method for solving ODEs based on the semi-implicit basic method of order 2. Considering several chaotic systems and van der Pol nonlinear oscillator as examples, we implemented a performance analysis of the proposed technique in comparison with well-known multistep methods: Adams–Bashforth, Adams–Moulton and the backward differentiation formula. We explicitly show that the multistep semi-implicit methods can outperform the classical linear multistep methods, providing more precision in the solutions for nonlinear differential equations. The analysis of stability regions reveals that the proposed methods are more stable than explicit linear multistep methods. The possible applications of the developed ODE solver are the long-term simulations of chaotic systems and processes, solving moderately stiff differential equations and advanced modeling systems. [ABSTRACT FROM AUTHOR]

We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation uzz [?] Lu = 0 in three space dimensions (x, y, z), where the domain is cylindrical in z. Cauchy data are given on the lower boundary and the boundary values on the upper boundary are sought. The problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of the two-dimensional elliptic operator L (via its eigenfunction expansion), and it is shown that the solution is stabilized (regularized) if the large eigenvalues are cut off. We suggest a numerical procedure based on the rational Krylov method, where the solution is projected onto a subspace generated using the operator L[?]1. This means that in each Krylov step, a well-posed two-dimensional elliptic problem involving L is solved. Furthermore, the hyperbolic cosine is evaluated explicitly only for a small symmetric matrix. A stopping criterion for the Krylov recursion is suggested based on the relative change of an approximate residual, which can be computed very cheaply. Two numerical examples are given that demonstrate the accuracy of the method and the efficiency of the stopping criterion. [ABSTRACT FROM AUTHOR]