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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.

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.

Biological materials (orthotropic materials), like wood, can offer good mechanical properties with a minimum amount of material, making their internal structure the suitable one to be applied on bio-inspired structures. The knowledge of the exceptional structural performance of palm trees, and specially its response to different loading conditions, provides useful information when lightweight structures with high slenderness ratio are desired. Recent researches focused on the analysis of palm trees subject to static loading conditions, ignoring the fluctuating nature of the wind speed. The purpose of this study is to simulate in a computational efficient way the effect of dynamic loading conditions applied on palm trees. Using the mode displacement method, the number of degrees of freedom of a dynamic finite element analysis can be drastically reduced with a minimal loss of accuracy. It was applied to simulate the behavior of structures comprised of an orthotropic material subject to a stochastic dynamic load. The influence of the number of selected degrees of freedom has also been studied. In addition, an exponential integration method is proposed to perform the time integration procedure. The results obtained show that a properly reduced model suitably represents the full finite element model without any appreciable loss of accuracy; it is also shown that computational cost can be drastically reduced. This method could give an appropriate computational representation of the behavior of orthotropic structures, and it could be used for studying more complex bio-inspired structures.

We present an approach for integrating the time evolution of quantum systems. We leverage the computation power of graphics processing units (GPUs) to perform the integration of all time steps in parallel. The performance boost is especially prominent for small to medium-sized quantum systems. The devised algorithm can largely be implemented using the recently-specified batched versions of the BLAS routines, and can therefore be easily ported to a variety of platforms. Our PARAllelized Matrix Exponentiation for Numerical Time evolution (PARAMENT) implementation runs on CUDA-enabled graphics processing units. Program summary Program Title: PARAMENT CPC Library link to program files: https://doi.org/10.17632/zy5v4xs89d.1 Developer's repository link: https://github.com/parament-integrator/parament Licensing provisions: Apache 2.0 Programming language: C / CUDA / Python Nature of problem: Time-integration of the Schrödinger equation with a time-dependent Hamiltonian for quantum systems with a small Hilbert space but many time-steps. Solution method: A 4th order Magnus integrator, highly parallelized on a GPU, implemented using a small subset of BLAS functions for improved portability., Computer Physics Communications, 270, ISSN:0010-4655, ISSN:1879-2944

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

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.

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.

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.

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

In this paper, we present new numerical methods to solve ordinary differential equations in both linear and nonlinear cases. we apply Daftardar-Gejji technique on theta-method to derive anew family of numerical method. It is shown that the method may be formulated in an equivalent way as a RungeKutta method. The stability of the methods is analyzed.

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.

In this paper, we present the Taylor polynomial solutions of system of higher order linear integro-differential Volterra-Fredholm equations (IDVFE). This method transforms IDVFE into the matrix equations which correspond to a system of linear algebraic equations. Some numerical results are also given to illustrate the efficiency of the method. Keywords: Taylor polynomial, Integro-differential equation, Volterra-Fredholm equations, Convergence analysis

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 .

Recently a splitting approach has been presented for the simulation of sonic-boom propagation. Splitting methods allow one to divide complicated partial differential equations into simpler parts that are solved by specifically tailored numerical schemes. The present work proposes a second order exponential integrator for the numerical solution of sonic-boom propagation modelled through a dispersive equation with Burgers' nonlinearity. The linear terms are efficiently solved in frequency space through FFT, while the nonlinear terms are efficiently solved by a WENO scheme. The numerical method is designed to be highly parallelisable and therefore takes full advantage of modern computer hardware. The new approach also improves the accuracy compared to the splitting method and it reduces oscillations. The enclosed numerical results illustrate that parallelisation on a CPU results in a speedup of 22 times faster than the straightforward sequential version. The GPU implementation further accelerates the runtime by a factor 3, which improves to 5 when single precision is used instead of double precision.

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