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In this paper, we propose two new exponential integrator Fourier pseudo-spectral schemes for nonlinear Dirac (NLD) equation. The proposed schemes are time symmetric, unconditionally stable and preserve the total energy in the discrete level. We give rigorously error analysis and establish error bounds in the general H m -norm for the numerical solutions of the new schemes applied to the NLD equation. In more details, the proposed schemes have the second-order temporal accuracy and spectral spatial accuracy, respectively, without any CFL-type condition constraint. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity to bound the numerical approximate solutions or the mathematical induction. Extensive numerical results are reported to confirm our error bounds and theoretical analysis.

The paper aims to develop a class of high-order explicit exponential time differencing energy-preserving schemes for some conservative fractional PDEs based on the general Hamiltonian form of these equations. The equation is first reformulated into an equivalent system that possesses a new quadratic energy by introducing an auxiliary variable. Then, explicit schemes are derived via applying the exponential time differencing Runge-Kutta method approximations for time integration and Fourier pseudo-spectral discretization in space, which can maintain the accuracy and improve the stability to the system with stiffness. Subsequently, highly efficient energy-preserving schemes are given by combining the proposed explicit schemes and the projection technique. Finally, the advantages of the proposed schemes are illustrated by the numerical results of the fractional nonlinear Schrodinger equation.

Recently, an exponential integrator Fourier pseudo-spectral (EIFP) scheme for the Klein–Gordon–Dirac (KGD) equation in the nonrelativistic limit regime has been proposed (Yi et al., 2019). The scheme is fully explicit and numerical experiments show that it is very efficient due to the fast Fourier transform (FFT). However, the authors did not give a strict convergence analysis and error estimate for the scheme. In addition, the scheme did not satisfy time symmetry which is an important characteristic of the exact solution. In this paper, by setting two-level format for Klein–Gordon part and three-level format for Dirac part, respectively, we proposed a new EIFP scheme for the KGD equation with periodic boundary conditions. The new scheme is time symmetric and fully explicit. By using the standard energy method and the mathematical induction, we make a rigorously convergence analysis and establish error estimates without any CFL condition restrictions on the grid ratio. The convergence rate of the proposed scheme is proved to be at the second-order in time and spectral-order in space, respectively, in a generic H m -norm. The numerical experiments are carried out to confirm our theoretical analysis. Because that our error estimates are given under the general H m -norm, the conclusion can easily be extended to two- and three-dimensional problems without the stability (or CFL) condition under sufficient regular conditions.

In this paper, based on the exponential integrator, a new Jacobi-type iteration method is proposed for solving linear system $$Ax=b$$ . The traditional Jacobi iteration method can be viewed as a special case of the new method. The convergence and two comparison theorems of the new Jacobi-type method are established for linear system with different type of coefficient matrices. The convergence of the traditional Jacobi iteration method follows immediately from these results. It is shown that for the linear system with coefficient matrix that is M-matrix or nonnegative matrix, the new method is convergent. Under suitable conditions, the spectral radius of iteration matrix for new method is much smaller than traditional Jacobi method for the case of nonnegative coefficient matrix. Numerical experiments are carried out to show the effectiveness of the new method when dealing with the nonnegative matrix system.

In order to describe quantum mechanical effects, the use of the von-Neumann equation is apparent. In this work, we present a unified numerical framework so that the von-Neumann equation in center-of-mass coordinates leads to a Quantum Liouville-type equation when choosing a suitable basis. In particular, the proposed approach can be related to the conventional Wigner equation when a plane wave basis is used. The drawback of the numerical methods is the high computational cost. Our presented approach is extended to allow reducing the dimension of the basis, which leads to a computationally efficient and accurate subdomain approach. Not only the steady-state behavior is of interest, but also the dynamic behavior. In order to solve the time-dependent case, suitable approximation methods for the time-dependent exponential integrator are necessary. For this purpose, we also investigate approximations of the exponential integrator based on Faber polynomials and Krylov methods. In order to evaluate and justify our approach, various test cases, including a resonant tunnel diode as well as a double-gate field-effect transistor, are investigated and validated for the stationary and the dynamic device behavior., Journal of computational electronics;Vol. 20. 2022, Issue 6, pp 2070-2090

In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided

In previous works we developed a methodology of deriving variational integrators to provide numerical solutions of systems having oscillatory behavior. These schemes use exponential functions to approximate the intermediate configurations and velocities, which are then placed into the discrete Lagrangian function characterizing the physical system. We afterwards proved that, higher order schemes can be obtained through the corresponding discrete Euler–Lagrange equations and the definition of a weighted sum of “continuous intermediate Lagrangians” each of them evaluated at an intermediate time node. In the present article, we extend these methods so as to include Lagrangians of split potential systems, namely, to address cases when the potential function can be decomposed into several components. Rather than using many intermediate points for the complete Lagrangian, in this work we introduce different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials. Finally, we assess the accuracy, convergence and computational time of the proposed technique by testing and comparing them with well known standards.

A Deuflhard-type exponential integrator sine pseudospectral (DEI-SP) method is proposed and analyzed for solving the generalized improved Boussinesq (GIBq) equation. The numerical scheme is based on a second-order exponential integrator for time integration and a sine pseudospectral discretization in space. Rigorous analysis and abundant experiments show that the method converges quadratically and spectrally in time and space, respectively. Finally the DEI-SP method is applied to investigate the complicated and interesting long-time dynamics of the GIBq equation.

A uniformly second order accurate nested Picard iterative integrator (NPI) is proposed to solve a system of oscillatory ordinary equations involving a parameter $$0

Linear and nonlinear evolution equations have been formulated to address problems in various fields of science and technology. Recently, methods using an exponential integrator for solving evolution equations, where matrix functions must be computed repeatedly, have been investigated and refined. In this paper, we propose a new method for computing these matrix functions which is called an inexact rational Krylov method. This is a more efficient version of the rational Krylov method with appropriate shifts, which was proposed by Hashimoto and Nodera (ANZIAM J 58:C149–C161, 2016). The advantage of the inexact rational Krylov method is that it computes linear equations that appear in the rational Krylov method efficiently while guaranteeing the accuracy of the final solution.

Faber polynomial based approximations of time dependent exponential integrators for the numerical solution of Maxwell's equations based on explicit algorithms are extended for the inclusion of nonlinear media so that multiphysical problems can be considered. For this purpose, appropriate exponential integrators, such as Lawson type and Rosenbrock type integrators are reviewed and investigated. The formulation with regard to Rosenbrock type integrators is extended to develop exponential integrators allowing the straightforward implementation of the resulting explicit algorithm and the application of large time step sizes opening up the potential for a parallel implementation. The Faber polynomial based approach for the approximation of the exponential integrator is utilized and as in the case of linear material models has an order of magnitude higher efficiency than comparable conventional methods, especially, when a high accuracy is required.

In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [ 21 ], with a high order RKEI method [ 7 ], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.

In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac {1}{2}$ H > 1 2 and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order $\mathcal {O}(h^{2}+\varDelta t)$ O ( h 2 + Δ t ) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

International audience; In this work, we focus on the numerical approximation of a hybrid fluid-kinetic plasma model for electrons, in which energetic electrons are described by a Vlasov kinetic model whereas a fluid model is used for the cold population of electrons. First, we study the validity of this hybrid modelling in a two dimensional context (one dimension in space and one dimension in velocity) against the full (stiff) Vlasov kinetic model and second, a four dimensional configuration is considered(one dimension in space and three dimensions in velocity) following [1]. To do so, we consider two numerical Eulerian methods. The first one is based on the Hamiltonian structure of the hybrid system and the second approach, which is based on exponential integrators, enables to derive high order integrator and remove the CFL condition induced by the linear part. The efficiency of these methods, which are combined with an adaptive time stepping strategy, are discussed in the different configurations and in the linear and nonlinear regimes.

Physics-based simulation methods for deformable objects suffer limitations due to the conflicting requirements that are placed on them. The work horse semi-implicit (SI) backward Euler method is very stable and inexpensive, but it is also a blunt instrument. It applies heavy damping, which depends on the timestep, to all solution modes and not just to high-frequency ones. As such, it makes simulations less lively, potentially missing important animation details. At the other end of the scale, exponential methods (exponential Rosenbrock Euler (ERE)) are known to deliver good approximations to all modes, but they get prohibitively expensive and less stable for very stiff material. In this article, we devise a hybrid, semi-implicit method called SIERE that allows the previous methods SI and ERE to each perform what they are good at. To do this, we employ at each timestep a partial spectral decomposition, which picks the lower, leading modes, applying ERE in the corresponding subspace. The rest is handled (i.e., effectively damped out) by SI. No solution of nonlinear algebraic equations is required throughout the algorithm. We show that the resulting method produces simulations that are visually as good as those of the exponential method at a computational price that does not increase with stiffness, while displaying stability and damping with respect to the high-frequency modes. Furthermore, the phenomenon of occasional divergence of SI is avoided.

In this paper, the numerical method for semi-linear stochastic variable delay integro-differential equations is studied. The stability of analytic solutions of semi-linear stochastic variable delay...

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schrödinger equation are included.

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha ,2}$ for some $\alpha \in (0,1)$. Due to the loss of smoothness in the problem, classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $\alpha +1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.

Ginzburg-Landau equation has a rich record of success in describing a vast variety of nonlinear phenomena such as liquid crystals, superfluidity, Bose-Einstein condensation and superconductivity to mention a few. Fractional order equations provide an interesting bridge between the diffusion wave equation of mathematical physics and intuition generation, it is of interest to see if a similar generalization to fractional order can be useful here. Non-integer order partial differential equations describing the chaotic and spatiotemporal patterning of fractional Ginzburg-Landau problems, mostly defined on simple geometries like triangular domains, are considered in this paper. We realized through numerical experiments that the Ginzburg-Landau equation world is bounded between the limits where new phenomena and scenarios evolve, such as sink and source solutions (spiral patterns in 2D and filament-like structures in 3D), various core and wave instabilities, absolute instability versus nonlinear convective cases, competition and interaction between sources and chaos spatiotemporal states. For the numerical simulation of these kind of problems, spectral methods provide a fast and efficient approach.

Motivated by the Hodgkin-Huxley model of neuronal dynamics, we study explicit numerical integrators for "conditionally linear" systems of ordinary differential equations. We show that splitting and composition methods, when applied to the Van der Pol oscillator and to the Hodgkin-Huxley model, do a better job of preserving limit cycles of these systems for large time steps, compared with the "Euler-type" methods (including Euler's method, exponential Euler, and semi-implicit Euler) commonly used in computational neuroscience, with no increase in computational cost. These limit cycles are important to preserve, due to their role in neuronal spiking. Splitting methods even compare favorably to the explicit exponential midpoint method, which is twice as expensive per step. The second-order Strang splitting method is seen to perform especially well across a range of non-stiff and stiff dynamics., 27 pages, 10 figures; v2: added discussion and analysis of exponential midpoint method, additional numerical experiments, remarks on reduced Hodgkin-Huxley models, and various minor revisions

Numerical time propagation of semi-linear equations of the Schrodinger type can be performed by the use of exponential integrators. The main difficulty for efficient implementation of this type of schemes lies in the evaluation of $$\varphi $$-functions of a matrix argument. We develop a Chebyshev series approximation for these functions and propose a simple algorithm for the evaluation of the series coefficients. The domain of convergence of the series is consistent with the spectrum of Schrodinger type operators. This approximation is shown to be accurate and performs favorably in comparison to other state of the art methods for approximation of $$\varphi $$-functions.

In this paper, we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods. It is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or Lyapunov functions. We also consider order estimates of the new methods. Furthermore, we explore and discuss the application of our methods in important stiff gradient systems, and it turns out that our methods are unconditionally energy-diminishing and strongly damped even for very stiff gradient systems. Practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear Schrodinger equation. As a byproduct of this paper, arbitrary-order trigonometric/RKN collocation methods are also presented and analysed for second-order highly oscillatory/general systems. The paper is accompanied by numerical results that demonstrate the great potential of this work.

It is well known that various dynamical systems preserve volume in phase space such as all Hamiltonian systems. This qualitative geometrical property of the analytical solution should be respected in the sense of Geometric Integration. This paper studies the volume-preserving property of exponential integrators for different vector fields. For exponential integrators, we first derive a necessary and sufficient condition of volume preservation. Then based on this condition, volume-preserving exponential integrators are discussed in detail for four kinds of vector fields. It is shown that symplectic exponential integrators can be volume preserving for a much larger class of vector fields than Hamiltonian systems. On the basis of the analysis, some applications of volume-preserving exponential integrators are discussed. For solving highly oscillatory second-order systems, novel volume-preserving exponential integrators are derived, and for separable partitioned systems, extended Runge–Kutta–Nystrom (ERKN) integrators of volume preservation are presented. Moreover, the volume preservation of Runge–Kutta–Nystrom (RKN) methods is also discussed. Five illustrative numerical experiments are carried out to demonstrate the notable superiority of volume-preserving exponential integrators in comparison with volume-preserving Runge-Kutta methods.

We re-visit the semi-Lagrangian Runge–Kutta exponential integrators presented in Celledoni et al. (J Sci Comput 41:139–164, 2009) for Convection-dominated convection-diffusion problems. Third order accurate methods of this class consist of subsets of coefficients of commutator-free exponential integrators including some free parameters. We consider an optimal choice of parameters for the third order accurate methods based on the least-squares methods applied on fourth order commutator-free conditions.

Exponential integrator (EI) method based on Krylov subspace approximation is a promising method for large-scale transient circuit simulation. However, it suffers from the singularity problem and consumes large subspace dimensions for stiff circuits when using the ordinary Krylov subspace. Restarting schemes are commonly applied to reduce the subspace dimension, but they also slow down the convergence and degrade the overall computational efficiency. In this paper, we first devise an implicit and sparsity-preserving regularization technique to tackle the singularity problem facing EI in the ordinary Krylov subspace framework. Next, we analyze the root cause of the slow convergence of the ordinary Krylov subspace methods when applied to stiff circuits. Based on the analysis, we propose a deflated restarting scheme, compatible with the above regularization technique, to accelerate the convergence of restarted Krylov subspace approximation for EI methods. Numerical experiments demonstrate the effectiveness of the proposed regularization technique, and up to 50% convergence improvements for Krylov subspace approximation compared to the non-deflated version.

In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.

Exponential integrators based on discrete gradient methods are applied to non-canonical Hamiltonian systems with added linear forcing/damping terms, which may be time-dependent. Changes in the dynamics, such as conservation of energy or Casimirs, which result from inclusion of the linear forcing/damping terms, are not exactly preserved by standard discrete gradient methods. However, those changes are shown to be exactly preserved by the exponential integrators in special circumstances. The methods are also symmetric, second order, and linearly stable. To demonstrate advantages in both accuracy and efficiency over other standard methods, the exponential integrators are applied to a three dimensional Lotka-Volterra system and a damped/driven Ablowitz-Ladik system.

In this paper we propose a fast compact exponential time differencing method for solving a class of semilinear parabolic equations with Neumann boundary conditions. The model equation is first discretized in space by a fourth-order compact finite difference scheme with an appropriate treatment of the boundary condition, the resulting semi-discretized system is then diagonalized with fast Fourier transforms, and further expressed in a temporal integral formulation by the use of exponential integrators according to the Duhamel principle. The fully discrete scheme is finally obtained by using multistep interpolations for the nonlinear terms and exact evaluations of the underlying integrals. Some numerical experiments are performed to demonstrate the accuracy and efficiency of the proposed method.

We introduce two exponential-type integrators for the “good” Boussinesq equation. They are of orders one and two, respectively, and they require lower spatial regularity of the solution compared to classical exponential integrators. For the first integrator, we prove first-order convergence in $$H^r$$ for solutions in $$H^{r+1}$$ with $$r>1/2$$ . This new integrator even converges (with lower order) in $$H^r$$ for solutions in $$H^r$$ , i.e., without any additional smoothness assumptions. For the second integrator, we prove second-order convergence in $$H^r$$ for solutions in $$H^{r+3}$$ with $$r>1/2$$ and convergence in $$L^2$$ for solutions in $$H^3$$ . Numerical results are reported to illustrate the established error estimates. The experiments clearly demonstrate that the new exponential-type integrators are favorable over classical exponential integrators for initial data with low regularity.

The paper presents error bounds for the second order exponential Runge-Kutta method for parabolic abstract linear time-dependent differential equations incorporating non-smooth initial data. As an example for this particular type of problems, the paper presents a spatial discretization of a partial integro-differential equation arising in financial mathematics, where non-smooth initial conditions occur in option pricing models. For this example, numerical studies of the convergence rate are given

High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approximation of exponential integrators (REXI). This method replaces a time integration of stiff linear oscillatory and diffusive systems by the sum of the solutions of many decoupled systems, which can be solved in parallel. Previous numerical studies showed that this reformulation allows taking arbitrarily long time steps for the linear oscillatory parts. The present work studies the non-linear shallow-water equations on the rotating sphere, a simplified system of equations used to study properties of space and time discretization methods in the context of atmospheric simulations. After introducing time integrators, we first compare the time step sizes to the errors in the simulation, discussing pros and cons of different formulations of REXI. Here, REXI already shows superior properties compared to explicit and implicit time stepping methods. Additionally, we present wallclock-time-to-error results revealing the sweet spots of REXI obtaining either an over 6 × higher accuracy within the same time frame or an about 3 × reduced time-to-solution for a similar error threshold. Our results motivate further explorations of REXI for operational weather/climate systems.

Motivated by the recent paper (Wang and Wu, 2018 [12] ), where the authors proposed a class of functionally-fitted energy-preserving integrators for Poisson systems, in this paper we aim at developing exponential integrators with quadratic energy preservation for the important class of linear Poisson systems. By rewriting the linear Poisson system in a new formulation, we apply the symplectic exponential Runge-Kutta methods and thus obtain a novel class of exponential integrators for the linear Poisson system. It is proved that the obtained exponential integrator can exactly preserve the quadratic energy. In comparison with the functionally-fitted energy-preserving methods whose numerical performance depends on the fitted frequency, the new exponential integrators are independent of the fitted frequency and convenient to use. Furthermore, we investigate the convergence of the fixed-point iteration used in the implementation of the proposed exponential integrator, and show that the convergence region of the stepsize is independent of the norm of the skew-symmetric matrix. Finally, numerical results with the Euler equations of a free rigid body soundly support the remarkable performance of the exponential integrator in both the global errors and the energy preservation.

This paper introduces an efficient unconditionally stable fourth-order method for solving nonlinear space-fractional reaction-diffusion systems with nonhomogeneous Dirichlet boundary conditions on bounded domains. The proposed method is based on a compact improved matrix transformed technique for fourth-order spatial approximation and exponential time differencing approximation for fourth-order time integration. The main advantage of the improved matrix transfer technique is that it leads to a system of ordinary differential equations with spatial discretization matrix raised to the desired fractional order. The key benefit of the fourth-order exponential integrator is that it can be implemented with essentially the same computational complexity as the backward Euler method by utilizing a partial fraction splitting technique in which it is just required to solve two backward Euler-type well-conditioned linear systems at each time step by computing LU decomposition of spatial discretization matrix once outside the time loop. Linear stability analysis and various numerical experiments are also performed to demonstrate stability and accuracy of the proposed method. Moreover, calculation of local truncation error and an empirical convergence analysis show the fourth-order accuracy of the proposed method.

This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena. The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems. We prove the strong convergence of the fully discrete schemes toward the mild solution. The results reveal how convergence orders depend on the regularity of the noise and the initial data. In addition, we exceed the classical orders 1 ∕ 2 in time and 1 in space achieved in the literature when dealing with SPDE driven by Poisson measure with less regularity assumptions on the nonlinear drift function. In particular, for trace class multiplicative Gaussian noise we achieve convergence order O ( h 2 + Δ t 1 ∕ 2 ) . For additive trace class Gaussian noise and an appropriate jump function, we achieve convergence order O ( h 2 + Δ t ) . Numerical experiments to sustain the theoretical results are provided.

The Exponential Integrator with Schur–Krylov Approximation (EISKA) algorithm was developed for combustion applications. This algorithm combines the advantages of the explicit large step advancement of the exponential schemes and the dimension reduction effect of the Krylov subspace approximation, and was improved by introducing the Schur decomposition to control the rounding error. The EISKA based on the SpeedCHEM (SC) package was implemented to simulate a methane partially stirred reactor (PaSR) with pair-wise mixing model by considering the mechanisms of Li et al., GRI-Mech 3.0 and USC Mech II. Accuracy and computational efficiency of EISKA are systematically compared with those of DVODE. In the case of the Li mechanism which is a priori sufficiently small to be handled directly in combustion simulations, the computations were accelerated by a factor of 1.99 without losing accuracy. In the cases of GRI-Mech 3.0 and USC Mech II which are significantly larger than the Li mechanism, chemical reduction methods, namely the Correlated Dynamic Adaptive Chemistry (CoDAC) and the Multi-timescale (MTS) method were coupled with either DVODE or EISKA. The results show that the EISKA is faster than DVODE either with or without chemical reduction methods. Model results show that the best strategy is to use EISKA without any reduction method which leads to the same accuracy as compared to DVODE and acceleration factors of 2.61 and 2.19 for GRI-Mech 3.0 and USC Mech II, respectively.

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.

In this paper, we consider exponential integrators for semilinear Poisson systems. Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs are presented. These geometric exponential integrators exhibit better long time stability properties as compared to non-geometric integrators, and are computationally more efficient than traditional symplectic integrators and energy-preserving methods based on the discrete gradient method., Comment: 18 pages, 11 figures

We consider the numerical approximation of the general second order semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space–time noise. Our goal is to build two numerical algorithms with strong convergence rates higher than that of the standard semi-implicit scheme. In contrast to the standard time stepping methods which use basic increments of the noise, we introduce two schemes based on the exponential integrators, designed for finite element, finite volume or finite difference space discretisations. We prove the convergence in the root mean square L 2 norm for a general advection diffusion reaction equation and a family of new Lipschitz nonlinearities. We observe from both the analysis and numerics that the proposed schemes have better convergence properties than the current standard semi-implicit scheme.

We study high-order Magnus-type exponential integrators for large systems of ordinary differential equations defined by a time-dependent skew-Hermitian matrix. We construct and analyze defect-based local error estimators as the basis for adaptive stepsize selection. The resulting procedures provide a posteriori information on the local error and hence enable the accurate, efficient, and reliable time integration of the model equations. The theoretical results are illustrated on two numerical examples .

In this paper we propose a novel way to integrate time-evolving partial differential equations that contain nonlinear advection and stiff linear operators, combining exponential integration techniques and semi-Lagrangian methods. The general formulation is built from the solution of an integration factor problem with respect to the problem written with a material derivative, so that the exponential integration scheme naturally incorporates the nonlinear advection. Semi-Lagrangian techniques are used to treat the dependence of the exponential integrator on the flow trajectories. The formulation is general, as many exponential integration techniques could be combined with different semi-Lagrangian methods. This formulation allows an accurate solution of the linear stiff operator, a property inherited by the exponential integration technique. It also provides a sufficiently accurate representation of the nonlinear advection, even with large time-step sizes, a property inherited by the semi-Lagrangian method. Aiming for application in weather and climate modeling, we discuss possible combinations of well established exponential integration techniques and state-of-the-art semi-Lagrangian methods used operationally in the application. We show experiments for the planar rotating shallow water equations. When compared to traditional exponential integration techniques, the experiments reveal that the coupling with semi-Lagrangian allows stabler integration with larger time-step sizes. From the application perspective, which already uses semi-Lagrangian methods, the exponential treatment could improve the solution of wave-dispersion when compared to semi-implicit schemes.

This paper presents a new algorithm KIOPS for computing linear combinations of φ -functions that appear in exponential integrators. This algorithm is suitable for large-scale problems in computational physics where little or no information about the spectrum or norm of the Jacobian matrix is known a priori . We first show that such problems can be solved efficiently by computing a single exponential of a modified matrix. Then our approach is to compute an appropriate basis for the Krylov subspace using the incomplete orthogonalization procedure and project the matrix exponential on this subspace. We also present a novel adaptive procedure that significantly reduces the computational complexity of exponential integrators. Our numerical experiments demonstrate that KIOPS outperforms the current state-of-the-art adaptive Krylov algorithm phipm .

In this paper we introduce a class of second-order exponential schemes for the time integration of semilinear wave equations. They are constructed such that the established error bounds only depend on quantities obtained from a well-posedness result of a classical solution. To compensate missing regularity of the solution the proofs become considerably more involved compared to a standard error analysis. Key tools are appropriate filter functions as well as the integration-by-parts and summation-by-parts formulas. We include numerical examples to illustrate the advantage of the proposed methods.

Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. The traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems., Fixed a definition and other minor typos. Results remain unchanged

Two different Perfectly Matched Layer (PML) formulations with efficient pseudo-spectral numerical schemes are derived for the standard and non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral explicit exponential integrator scheme for a first-order formulation and a linearly implicit preconditioned finite-difference scheme for a second-order formulation are proposed and analyzed. To obtain a high spatial accuracy, new regularized Berm\'udez type absorption profiles are introduced for the PML. It is shown that the two schemes are efficient, but the linearly implicit scheme should be preferred for accuracy purpose when used within the framework of pseudo-spectral methods combined with the regularized Berm\'udez type functions. In addition, in the non-relativistic regime, numerical examples lead to the conclusion that the error related to regularized Berm\'udez type absorption functions is insensitive to the small parameter $\varepsilon$ involved in the NKGE. The paper ends by a two-dimensional example showing that the strategy extends to the rotating NKGE where the vortex dynamics is very well-reproduced., Comment: 26 pages

Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units., 25 pages, 5 figures, submitted to Journal of Computational Physics

Since various dynamical systems preserve volume in phase space, such as all Hamiltonian systems, this qualitative geometrical property of the analytical solution should be preserved within the framework of Geometric Integration. This chapter considers the volume-preserving exponential integrators for different vector fields. We first analyse a necessary and sufficient condition of volume preservation for exponential integrators. We then discuss volume-preserving exponential integrators for four kinds of vector fields. It turns out that symplectic exponential integrators can be volume preserving for a much larger class of vector fields than Hamiltonian systems. On the basis of this profound analysis, the applications of volume-preserving exponential integrators are demonstrated. For solving highly oscillatory second-order systems, efficient volume-preserving exponential integrators are derived, and for separable partitioned systems, volume-preserving ERKN integrators are presented. Moreover, volume-preserving RKN methods are also investigated.

Time integration of advection dominated advection–diffusion problems on refined meshes can be a challenging task, since local refinement can lead to a severe time step restriction, whereas standard implicit time stepping is usually hardly suitable for treating advection terms. We show that exponential time integrators can be an efficient, yet conceptually simple, option in this case. Our comparison includes three exponential integrators and one conventional scheme, the two-stage Rosenbrock method ROS2 which has been a popular alternative to splitting methods for solving advection–diffusion problems.