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We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschitz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the step size unlike the GBM tamed method. [ABSTRACT FROM AUTHOR]

In this article, we present a parareal exponential finite element method, with the help of variational formulation and parareal framework, for solving semilinear parabolic equations in rectangular domains. The model equation is first discretized in space using the finite element method with continuous piecewise multilinear rectangular basis functions, producing the semi‐discrete system. We then discretize the temporal direction using the explicit exponential Runge–Kutta approach accompanied by the parareal framework, resulting in the fully‐discrete numerical scheme. To further improve computational speed, we design a fast solver for our method based on tensor product spectral decomposition and fast Fourier transform. Under certain regularity assumption, we successfully derive optimal error estimates for the proposed parallel‐based method with respect to H1$$ {H}^1 $$‐norm. Extensive numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our method. [ABSTRACT FROM AUTHOR]

Scientific inquiry into effective numerical methods for modelling complex physical processes has led to the investigation of fluid dynamics, mainly when non-Newtonian properties and complex heat sources are involved. This paper presents an enhanced exponential time integrator approach to dynamically simulate non-Newtonian boundary layer flow with spatially and temporally varying heat sources. We propose an explicit scheme with second-order accuracy in time, demonstrated to be stable through Fourier series analysis, for solving time-dependent partial differential equations (PDEs). Utilizing this scheme, we construct and solve dimensionless PDEs representing the flow of Williamson fluid under the influence of space- and temperature-dependent heat sources. The scheme discretizes the continuity equation of incompressible fluid and Navier–Stokes, energy, and concentration equations using the central difference in space. Our analysis illuminates how factors affect velocity, temperature, and concentration profiles. Specifically, we observe a rise in temperature profile with enhanced coefficients of space and temperature terms in the heat source. Non-Newtonian behaviours and geographical/temporal variations in heat sources are critical factors influencing overall dynamics. The novelty of our work lies in developing an explicit exponential integrator approach, offering stability and second-order accuracy, for solving time-dependent PDEs in non-Newtonian boundary layer flow with variable heat sources. Our results provide valuable quantitative insights for understanding and controlling complex fluid dynamics phenomena. By addressing these challenges, our study advances numerical techniques for modelling real-world systems with implications for various engineering and scientific applications.

Exponential integrators have been applied successfully in several physics-related differential equations. However, their application in hyperbolic systems with absorbing boundaries, like the ones arising in seismic imaging, still lacks theoretical and experimental investigations. The present work conducts an in-depth study of exponential integration using Faber polynomials, consisting of a generalization of a well-known exponential method that uses Chebyshev polynomials. This allows solving non-symmetric operators that emerge from classic seismic wave propagation problems with absorbing boundaries. Theoretical as well as numerical results are presented for Faber approximations. One of the theoretical contributions is the proposal of a sharp bound for the approximation error of the exponential of a normal matrix. We also show the practical importance of determining an optimal ellipse encompassing the full spectrum of the discrete operator to ensure and enhance the convergence of the Faber exponential series. Furthermore, based on estimates of the spectrum of the discrete operator of the wave equations with a widely used absorbing boundary method, we numerically investigate the stability, dispersion, convergence, and computational efficiency of the Faber exponential scheme. Overall, we conclude that the method is suitable for seismic wave problems and can provide accurate results with large time step sizes, with computational efficiency increasing with the increase of the approximation degree. [ABSTRACT FROM AUTHOR]

In this work, we present a high-order numerical method for the two-dimensional nonlinear space-fractional complex Ginzburg-Landau equation (FCGLE). Firstly, a fourth-order approximation is adopted to discretize the spatial Riesz fractional derivatives that leads to a semi-linear system of ordinary differential equations (ODEs), whose coefficient matrix has the complex block Toeplitz structure. Then a fourth-order exponential integrator method is used to solve the corresponding semi-linear ODEs system. In light of the results in theory, the proposed algorithm is fourth-order accuracy in both time and space. In the specific implementation of the proposed algorithm, due to the special structure of the coefficient matrix, the products of some φ -functions of matrices (related to the matrix exponential) and vectors are computed by the shift-invert Lanczos technique in the exponential integrator. In order to calculate the linear system of equations arising from the shift-invert Lanczos procedure, two classes of efficient preconditioners including Strang's circulant preconditioner/ τ matrix preconditioner are constructed and implemented by fast Fourier transform and fast sine transform, respectively. Numerical examples with and without exact solutions are implemented to confirm the effectiveness of the current algorithm. [ABSTRACT FROM AUTHOR]

A new spectral method is constructed for the linear and semilinear subdiffusion equations with possibly discontinuous rough initial data. The new method effectively combines several computational techniques, including the contour integral representation of the solutions, the quadrature approximation of contour integrals, the exponential integrator using the de la Vall\'ee Poussin means of the source function, and a decomposition of the time interval geometrically refined towards the singularity of the solution and the source function. Rigorous error analysis shows that the proposed method has spectral convergence for the linear and semilinear subdiffusion equations with bounded measurable initial data and possibly singular source functions under the natural regularity of the solutions. [ABSTRACT FROM AUTHOR]

A large class of differential Riccati equations (DREs) satisfy positivity property in the sense that the time-dependent solution preserves for any time its symmetric and positive semidefinite structure. This positivity property plays a crucial role in understanding the wellposedness of the DRE, and whether it could be inherited in the discrete level is a significant issue in numerical simulations. In this paper, we study positivity preserving time integration schemes by means of exponential integrators. The proposed exponential Euler and exponential midpoint schemes are linear and proven to be positivity preserving and unconditionally stable. Sharp error estimates of the schemes are also obtained. Numerical experiments are carried out to illustrate the performance of the proposed integrators. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semidiscrete system to produce a fully discrete numerical solution. Under certain regularity assumptions, error estimates measured in H¹-norm are successfully derived for the proposed schemes with one and two Runge-Kutta stages. More remarkably, the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix, which provides a fast solution process based on tensor product spectral decomposition and the fast Fourier transform. Various numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the excellent performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the class of Jacobian-free High Order Local Linearization (HOLL) methods is introduced for integrating large systems of initial value problems. Unlike other high order exponential integrators, this new class of methods involves the approximation of just a single phi-function times vector and does not require the evaluation and storing of Jacobian matrices, which result more efficient in terms of flops operations and computer memory. Specifically, the new Jacobian-free integrators are constructed by approximating the products of Jacobian matrix times vector that appear in the ordinary HOLL methods. A general result on the convergence rate of the new methods is derived in terms of the convergence rate of the mentioned approximations, resulting in a simple order condition that preserves the order of the ordinary HOLL methods. To design effective integrators of the new class, a novel Matrix-free Krylov-Padé approximation to the product of phi-function times vector is also proposed as well as an adaptive strategy for the automatic selection of the Krylov dimension and Padé order. As a particular instance, a class of Jacobian-free Locally Linearized Runge-Kutta schemes is developed, and schemes of third to fifth order explicitly constructed. Numerical simulations are provided illustrating the theoretical findings concerning the convergence rate of the introduced methods as well as their performance in comparison with other Jacobian-free integrators. • Jacobian-free high order exponential integrator with a single phi-function times vector per integration step. • Only one Krylov subspace approximation and matrix exponential per integration step. • Convergence rate with a single order condition. • Computation of phi-functions times vectors with high order Matrix-free Krylov-Padé approximation. • Adaptive estimation of Krylov subspaces dimensions with low fluctuations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Two-component reaction–diffusion models in high dimensions involving space-fractional derivatives are used as a powerful modelling approach for understanding several aspects of spatial heterogeneity and nonlocality. In this paper, we propose an accurate, unconditionally stable, and maximum principle preserving fourth-order method both in space and time variables to study the complex dynamical processes of two-component nonlinear space-fractional reaction–diffusion systems posed in high dimensions. To achieve a fast fourth-order accurate method, we adapt the matrix transfer technique with a fast Fourier transform-based implementation in space and an exponential integrator in time. The main advantage of the method is that it avoids storing the large dense matrix resulting from discretizing the fractional operator with the matrix transform approach and significantly reduces the computational costs. Some numerical experiments are carried out in two and three space dimensions to demonstrate the accuracy and computational efficiency of the method. [ABSTRACT FROM AUTHOR]

Radiation‐induced photocurrent effects represent a threat to microelectronic components operating in space, manmade and terrestrial radiation environments. Analysis of these threats by circuit simulations requires accurate and computationally efficient compact models. Most existing compact models are based on closed form analytic solutions of the governing equations and require empirical assumptions and idealizations that can limit their validity. In this paper we formulate an alternative numerical, data‐driven approach that learns a compact model from data representative of the type of measurements one can obtain in an experimental facility. To develop the model we start from a generic discrete‐time dynamical system and then use physics knowledge to refine its structure. Numerical studies demonstrate the potential of the model and establish some empirical guidelines for its training. [ABSTRACT FROM AUTHOR]

Magnetophoresis is one of the most popular particle manipulation schemes used in microfluidic devices. Predicting the performance of arbitrary magnetophoretic module via simulation is crucial for the realization of high-performance, purpose-built magnetophoretic devices. However, the simulation process may take a long time, slowing down the simulation throughput. This work presents a novel algorithm to rapidly predict the trajectory of magnetic particles in microfluidic devices with arbitrary magnetophoretic structures under continuous fluidic flow conditions. Based on the Eulerian–Lagrangian approach, by employing numerically calculated magnetic force and fluidic velocity datasets obtained by finite element analysis software, we used an exponential integrator inspired numerical method to solve for the motion of magnetic particles in the system with the assumption that the magnetic and fluidic forces are constants within a small period of time. The results showed that, for our application, the order of truncation error of our algorithm is comparable with the 4th Order Runge–Kutta method and commercial Runge–Kutta based solver ODE45 from MATLAB. Furthermore, our algorithm is inherently stable, thus enabling the use of arbitrary time stepping values without the need of adaptive step size adjustment, which significantly reduces simulation time when compared to the 4th Order Runge–Kutta method and ODE45. In our simulations, a maximum of 40 times reduction in simulation time can be achieved. The proposed algorithm and the performed simulations are validated experimentally. [ABSTRACT FROM AUTHOR]

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen--Cahn equation. While some commonly used first-order time stepping schemes have turned out to preserve unconditionally both the energy dissipation law and the MBP for the equation, restrictions on the time step size are still needed for existing secondorder or even higher order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen--Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but we also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we develop a framework to construct energy-preserving methods for multi-component Hamiltonian systems, combining the exponential integrator and the partitioned averaged vector field method. This leads to numerical schemes with advantages of original energy conservation, long-time stability and excellent behavior for highly oscillatory or stiff problems. Compared to the existing energy-preserving exponential integrators (EP-EI) in practical implementation, our proposed methods are much efficient which can at least be computed by subsystem instead of handling a nonlinear coupling system at a time. Moreover, for most cases, such as the Klein-Gordon-Schrödinger equations and the Klein-Gordon-Zakharov equations considered in this paper, the computational cost can be further reduced. Specifically, one part of the derived schemes is totally explicit and the other is linearly implicit. In addition, we present rigorous proof of conserving the original energy of Hamiltonian systems, in which an alternative technique is utilized so that no additional assumptions are required, in contrast to the proof strategies used for the existing EP-EI. Numerical experiments are provided to demonstrate the significant advantages in accuracy, computational efficiency and the ability to capture highly oscillatory solutions. [ABSTRACT FROM AUTHOR]

The fractional differential equation has been used to describe many phenomenons in almost all applied sciences, such as fluid flow in porous materials, anomalous diffusion transport, acoustic wave propagation in viscoelastic materials, and others. In some cases, it is complicated to find the analytical solution of a fractional differential equation, so there are some special techniques to approximate the solution. In this paper, a new collocation method for a class of semilinear fractional differential equations has been constructed. The convergence properties of the proposed methods and the computational complexity are treated. Some numerical experiments are presented to validate the theoretical results. The proposed numerical method provides an alternative way to study some physical phenomena by fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper introduces Jacobian-free Locally Linearized Runge–Kutta formulas of Dormand and Prince for integrating large systems of initial value problems. With these new Jacobian-free formulas, an adaptive time-stepping variable-order scheme with regulated Krylov dimension is constructed with new developments in the computation of the phi-function action over vectors through Jacobian-free Krylov–Padé approximations, and in the procedures for controlling the approximation errors and for estimating the dimension of the Krylov subspaces. At each integration step, the implemented scheme performs only one Krylov subspace decomposition and a few computations of low dimensional exponential matrices, which contrasts with the implementation of other exponential-type integrators of high order. Regarding to existing schemes, the performed numerical study demonstrates a higher effectiveness of the constructed scheme in the integration of a variety of test equations. • Jacobian-free Locally Linearized Runge–Kutta formulas of Dormand and Prince. • Simple order condition. • Adaptive Jacobian-free schemes with single Krylov decomposition per integration step. • Variable order Matrix-free Krylov-Padé approximation for phi-function times vectors. • Adaptive estimation of Krylov subspace dimensions with low fluctuations. [ABSTRACT FROM AUTHOR]

In this paper, second-order and fourth-order linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries (KdV) equations are proposed. First, the KdV equation is semi-discretized into an oscillatory Hamiltonian system of Ordinary Differential Equations (ODEs) by semi-discretizations of the skew adjoint operator and the Hamiltonian in the Hamiltonian form of the KdV equation. Then the resulting semi-discrete system is integrated by exponential average vector field integrator. The order of convergence and the conservation properties are established for the introduced full-discrete procedures. It shows that the proposed methods are energy-preserving, mass-preserving and qualitatively preserve the dispersive relation well. Numerical experiments are carried out to validate the efficiency and the accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method. • Multiscale reduction methods are efficient and accurate to solve problems with high-contrast media. • The presence of high-contrast coefficients reduces the stability region of time discretization. • Implicit methods must solve nonlinear equations, which can be a bottleneck computation. • Exponential integrators are alternative techniques and use robust time-stepping. [ABSTRACT FROM AUTHOR]

An exponential type of convolution quadrature is proposed as a time-stepping method for the nonlinear subdiffusion equation with bounded measurable initial data. The method combines contour integral representation of the solution, quadrature approximation of contour integrals, multistep exponential integrators for ordinary differential equations, and locally refined stepsizes to resolve the initial singularity. The proposed k-step exponential convolution quadrature can have kth-order convergence for bounded measurable solutions of the nonlinear subdiffusion equation based on naturalregularity of the solution with bounded measurable initial data. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, based on the exponential integrator, a new Jacobi-type iteration method is proposed for solving linear system A x = 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, an efficient numerical method is developed for solving a class of nonlinear fractional differential equations. The main idea is to transform the nonlinear fractional differential equations into a system of integral equations involved the generalized Mittag‐Leffler functions, and then to discretize the integral equations by the technique of exponential integrators and collocation methods with the polynomials of Lagrange basis. The convergence of this method is proven. The linear stability analysis of this method is carried out, and the stability region is derived. Finally, numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this study is to propose a combined numerical treatment for the dispersive partial differential equations involving dissipation, convection and reaction terms with nonlinearity, such as the KdV‐Burgers, KdV and dispersive‐Fisher equations. We use the combination of the exponential Rosenbrock–Euler time integrator and multiquadric‐radial basis function meshfree scheme in space as a qualitatively promising and computationally inexpensive method to efficiently exhibit behavior of such fruitful interactions resulting in antikink, two solitons and antikink‐breather waves. Obtained numerical solutions are compared with the existing results in the literature and discussed using illustrations in detail. [ABSTRACT FROM AUTHOR]

A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach, the exponential k-step method would have k th -order convergence in approximating a mild solution, possibly nonsmooth at the initial time. In consistency with the theoretical analysis, a numerical example shows that the method can achieve high-order convergence in the maximum norm for semilinear parabolic equations with discontinuous initial data. [ABSTRACT FROM AUTHOR]

A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method consists of a kth-order multistep exponential integrator in time and a lumped mass finite element method in space with piecewise rth-order polynomials and Gauss--Lobatto quadrature. At every time level, the extra values violating the maximum principle are eliminated at the finite element nodal points by a cut-off operation. The remaining values at the nodal points satisfy the maximum principle and are proved to be convergent with an error bound of O(Tk + hr). The accuracy can be made arbitrarily high-order by choosing large k and r. Extensive numerical results are provided to illustrate the accuracy of the proposed method and the effectiveness in capturing the pattern of phase-field problems. [ABSTRACT FROM AUTHOR]

We propose a Deuflhard-type exponential integrator Fourier pseudo-spectral (DEI-FP) method for solving the “Good” Boussinesq (GB) equation. The numerical scheme is based on a Deuflhard-type exponential integrator and a Fourier pseudo-spectral method for temporal and spatial discretizations, respectively. The scheme is fully explicit and efficient due to the fast Fourier transform. Rigorous error estimates are established for the method without any CFL-type condition constraint. In more details, the method converges quadratically and spectrally in time and space, respectively. Extensive numerical experiments are reported to confirm the theoretical analysis and to demonstrate rich dynamics of the GB equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose efficient and reliable numerical methods to solve two notable non-integer-order partial differential equations. The proposed algorithm adapts the Fourier spectral method in space, coupled with the exponential integrator scheme in time. As an advantage over existing methods, our method yields a full diagonal representation of the non-integer fractional operator, with better accuracy over a finite difference scheme. We realize in this work that evolution equations formulated in the form of fractional-in-space reaction-diffusion systems can result in some amazing examples of pattern formation. Numerical experiments are performed in two and three space dimensions to justify the theoretical results. Simulation results revealed that pattern formation in a fractional medium is practically the same as in classical reaction-diffusion scenarios.

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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take advantage of the exact solution of (generalized) geometric Brownian motion. We examine both Euler and Milstein versions of the scheme and prove strong convergence, taking care to deal with the dependence on the noise in the solution operator. For the special case of linear noise we obtain an improved rate of convergence for the Euler version over standard integration methods. We investigate the efficiency of the methods compared with other exponential integrators for low dimensional SDEs and high dimensional SDEs arising from the discretization of stochastic partial differential equations. We show that, by introducing a suitable homotopy parameter, these schemes are competitive not only when the noise is linear, but also in the presence of nonlinear noise terms. Although our new schemes are derived and analysed under zero commutator conditions (1.2), our numerical investigations illustrate that the resulting methods rival traditional methods even when this does not hold. [ABSTRACT FROM AUTHOR]

Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general formulation of exponential integrators is offered by the Exponential Propagation Iterative methods of Runge-Kutta type (EPIRK) family of schemes. The use of Jacobian approximations is an important strategy to drastically reduce the overall computational costs of implicit schemes while maintaining the quality of their solutions. This paper extends the EPIRK class to allow the use of inexact Jacobians as arguments of the matrix exponential-like functions. Specifically, we develop two new families of methods: EPIRK-W integrators that can accommodate any approximation of the Jacobian, and EPIRK-K integrators that rely on a specific Krylov-subspace projection of the exact Jacobian. Classical order conditions theories are constructed for these families. Practical EPIRK-W methods of order three and EPIRK-K methods of order four are developed. Numerical experiments indicate that the methods proposed herein are computationally favorable when compared to a representative state-of-the-art exponential integrator, and a Rosenbrock-Krylov integrator. [ABSTRACT FROM AUTHOR]

This paper presents, discusses and analyses a massively parallel-in-time solver for linear oscillatory partial differential equations, which is a key numerical component for evolving weather, ocean, climate and seismic models. The time parallelization in this solver allows us to significantly exceed the computing resources used by parallelization-in-space methods and results in a correspondingly significantly reduced wall-clock time. One of the major difficulties of achieving Exascale performance for weather prediction is that the strong scaling limit – the parallel performance for a fixed problem size with an increasing number of processors – saturates. A main avenue to circumvent this problem is to introduce new numerical techniques that take advantage of time parallelism. In this paper, we use a time-parallel approximation that retains the frequency information of oscillatory problems. This approximation is based on (a) reformulating the original problem into a large set of independent terms and (b) solving each of these terms independently of each other which can now be accomplished on a large number of high-performance computing resources. Our results are conducted on up to 3586 cores for problem sizes with the parallelization-in-space scalability limited already on a single node. We gain significant reductions in the time-to-solution of 118.3× for spectral methods and 1503.0× for finite-difference methods with the parallelization-in-time approach. A developed and calibrated performance model gives the scalability limitations a priori for this new approach and allows us to extrapolate the performance of the method towards large-scale systems. This work has the potential to contribute as a basic building block of parallelization-in-time approaches, with possible major implications in applied areas modelling oscillatory dominated problems. [ABSTRACT FROM AUTHOR]

We propose new explicit exponential Runge-Kutta methods for the weak approximation of solutions of stiff Itô stochastic differential equations (SDEs). We also consider the use of exponential Runge-Kutta methods in combination with splitting methods. These methods have weak order 2 for multidimensional, noncommutative SDEs with a semilinear drift term, whereas they are of order 2 or 3 for semilinear ordinary differential equations. These methods are A-stable in the mean square sense for a scalar linear test equation whose drift and diffusion terms have complex coefficients. We carry out numerical experiments to compare the performance of these methods with an existing explicit stabilized method of weak order 2. [ABSTRACT FROM AUTHOR]

We propose and analyse an exponential approach to solving highly ill-conditioned linear systems with a positive definite matrix. This approach also provides a new technique to find the inverse of a given positive definite matrix by a one-stop procedure. The advantages of this approach are its conceptual simplicity and ease of implementation. Numerical experiments are carried out. Numerical results demonstrate the effectiveness and superiority of the exponential approach to solving highly ill-conditioned systems of algebraic equations in comparison with the standard MATLAB codes. [ABSTRACT FROM AUTHOR]