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Exponential time differencing methods are a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models often possess fast oscillating or decaying modes—in other words, are stiff systems. Practical implementation of these methods for the systems with nondiagonal linear part of equations is exacerbated by infeasibility of an analytical calculation of the exponential of a nondiagonal linear operator; in this case, the coefficients of the exponential time differencing scheme cannot be calculated analytically. We suggest an approach, where these coefficients are numerically calculated with auxiliary problems. We rewrite the high-order Runge–Kutta type schemes in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for a heterogeneous Cahn–Hilliard equation, a sixth-order spatial derivative equation governing pattern formation in the presence of an additional conservation law, and a Fokker–Planck equation governing macroscopic dynamics of a network of neurons. • An approach to implementation of exponential time differencing methods is presented. • The approach allows for simulation of equation systems with nondiagonal linear part. • Performance gain for high-precision calculations is up to several orders of magnitude. • Efficient for PDEs, Fokker-Planck equations, and large ensembles of oscillators. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the use of higher-order exponential Rosenbrock time integration methods for the shallow water equations on the sphere. This stiff, nonlinear model provides a 'testing ground' for accurate and stable time integration methods in weather modeling, serving as the focus for exploration of novel methods for many years. We therefore identify a candidate set of three recent exponential Rosenbrock methods of orders four and five (exprb42, pexprb43 and exprb53) for use in this model. Based on their multi-stage structure, we propose a set of modifications to the phipm/IOM2 algorithm for efficiently calculating the matrix functions φ k. We then investigate the performance of these methods on a suite of four challenging test problems, comparing them against the epi3 method investigated previously in [1,2] on these problems. In all cases, the proposed methods enable accurate solutions at much longer time-steps than epi3, proving considerably more efficient as either the desired solution error decreases, or as the test problem nonlinearity increases. Highlights • A set of three exponential Rosenbrock methods is proposed for the SWEs on the sphere. • The new approach allows to approximate the nonlinearity with much increased accuracy. • Propose an improved code, phipm_simul_iom2, for implementing exponential integrators. • Enable accurate solutions at much longer time-steps that result in much faster run-times. [ABSTRACT FROM AUTHOR]

Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EPIRK-type are used to simulate magnetic reconnection and the Kevin–Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDF scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD. [ABSTRACT FROM AUTHOR]

Exponential propagation iterative (EPI) methods provide an efficient approach to the solution of large stiff systems of ODEs, compared to standard integrators. However, the bulk of the computational effort in these methods is due to products of matrix functions and vectors, which can become very costly at high resolution due to an increase in the number of Krylov projection steps needed to maintain accuracy. In this paper, it is proposed to modify EPI methods by using Krylov subspace spectral (KSS) methods, instead of standard Krylov projection methods, to compute products of matrix functions and vectors. Numerical experiments demonstrate that this modification causes the number of Krylov projection steps to become bounded independently of the grid size, thus dramatically improving efficiency and scalability. As a result, for each test problem featured, as the total number of grid points increases, the growth in computation time is just below linear, while other methods achieved this only on selected test problems or not at all. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the framework of the exponential propagation iterative methods of Runge–Kutta type (EPIRK) is extended to construct split EPIRK (sEPIRK) integrators for semilinear systems of ODEs. The structure of the sEPIRK methods possesses the flexibility and generality that allows construction of very efficient schemes. We demonstrate how bicolored trees-based B-series can be used to derive the order conditions for the new integrators. An algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed. The numerical results illustrate the advantages offered by the new class of integrators. The experiments also address the comparative performance of split vs. non-split EPIRK methods and the question of improving efficiency by optimizing coefficients of the sEPIRK schemes. It is shown that specific schemes can be custom-built to improve computational efficiency for a particular problem. [Copyright &y& Elsevier]

Abstract: We present a coupling of multiscale frameworks with accelerated stochastic simulation algorithms for systems of chemical reactions with disparate propensities. The algorithms regulate the propensities of the fast and slow reactions of the system, using alternating micro and macro sub-steps simulated with accelerated algorithms such as and R-leaping. The proposed algorithms are shown to provide significant speedups in simulations of stiff systems of chemical reactions with a trade-off in accuracy as controlled by a regulating parameter. More importantly, the error of the methods exhibits a cutoff phenomenon that allows for optimal parameter choices. Numerical experiments demonstrate that hybrid algorithms involving accelerated stochastic simulations can be, in certain cases, more accurate while faster, than their corresponding stochastic simulation algorithm counterparts. [Copyright &y& Elsevier]

Abstract: Stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems; it is not only important for a time-stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately. In the context of stochastic chemical systems, time stepping methods are known as tau leaping. Well known existing tau leaping methods have shortcomings in this regard. The implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance. On the other hand, the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems. We propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system . The proposed method uses a central limit approximation (CLA) locally over the tau leaping interval and is referred to as the LCLA-τ. The CLA predicts the mean and covariance as solutions of certain differential equations (ODEs) and for efficiency we solve these using a single time step of a suitable low order method. We perform a mean/covariance stability analysis of various possible low order schemes to determine the best scheme. Numerical experiments presented show that LCLA-τ performs favorably for stiff systems and that the LCLA-τ is also able to capture bimodal distributions unlike the CLA itself. The proposed LCLA-τ method uses a split implicit step to compute the mean update. We also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit Euler method as applied to the fluid limit differential equation. [Copyright &y& Elsevier]

Abstract: We propose a new class of the exponential propagation iterative methods of Runge–Kutta-type (EPIRK). The EPIRK schemes are exponential integrators that can be competitive with explicit and implicit methods for integration of large stiff systems of ODEs. Introducing the new, more general than previously proposed, ansatz for EPIRK schemes allows for more flexibility in deriving computationally efficient high-order integrators. Recent extension of the theory of B-series to exponential integrators is used to derive classical order conditions for schemes up to order five. An algorithm to systematically solve these conditions is presented and several new fifth order schemes are constructed. Several numerical examples are used to verify the order of the methods and to illustrate the performance of the new schemes. [Copyright &y& Elsevier]

Abstract: Numerical aspects of computational modeling of chemical vapor deposition are discussed. Large sparse strongly nonlinear algebraic systems are to be solved per time step. For this, inexact Newton methods and preconditioned Krylov subspace methods are suitable. To ensure positivity of concentrations, we propose a novel approach, namely a projected inexact Newton method. Unlike the commonly used method of clipping, this conserves mass. Efficiency of several preconditionings is compared. Our numerical tests culminate in an unusually large computation, namely a three-dimensional case with 17 species and 26 reactions. [Copyright &y& Elsevier]

We present the construction and preliminary tests of the exponential time integration schemes for the Euler equations. The equations are written in terms of potential temperature and Exner function. The time stepping is accomplished with the EPI2 and EPI3 schemes which have been used in the past for very accurate, long time step integration of the shallow water equations on the sphere. The stability of exponential integrators of this class is ensured by approximation of the oscillatory term using an absolutely stable exponential formula. Consequently, the schemes are not subject to the CFL condition imposed by the linear oscillatory part of the system and they also eliminate the phase errors associated with all the known implicit time stepping algorithms. All calculations are performed using the phipm algorithm based on the Krylov space methods. It is shown that the spectral radius of the Jacobian and consequently the Krylov space dimension are very sensitive to the presence of spatial noise. Efficiency and accuracy increase after the Shapiro filter is applied every two time steps. The method is investigated using several standardized and well accepted benchmark tests, including convective bubbles and a cold density current proposed by Straka and collaborators. Numerical experiments show that a stable integration is obtained for the wave Courant number of the order of 60 to 250. We present a detailed discussion of the development of a cold density current capturing the spiraling Prandtl vortex and the subsequent appearance of the Kelvin–Helmholtz billows. Our results from the exponential method compare very well with those obtained from the accurate schemes reported in the recent literature. The algorithm accurately handles both advection as well as acoustic and gravity waves. These results warrant the further development of the scheme for modeling atmospheric processes on a wide range of scales. • Exponential methods are applied to integrate Euler's equations for atmospheric convection. • The exponential time stepping eliminates the CFL restriction imposed by acoustic and gravity waves. • The calculations are performed using the PHIPM algorithm based on Krylov projection methods. • The numerical schemes are evaluated using convective tests and Straka density current. [ABSTRACT FROM AUTHOR]

Abstract: A new class of exponential propagation techniques which we call exponential propagation iterative (EPI) methods is introduced in this paper. It is demonstrated how for large stiff systems these schemes provide an efficient alternative to standard integrators for computing solutions over long time intervals. The EPI methods are constructed by reformulating the integral form of a solution to a nonlinear autonomous system of ODEs as an expansion in terms of products between special functions of matrices and vectors that can be efficiently approximated using Krylov subspace projections. The methodology for constructing EPI schemes is presented and their performance is illustrated using numerical examples and comparisons with standard explicit and implicit integrators. The history of the exponential propagation type integrators and their connection with EPI schemes are also discussed. [Copyright &y& Elsevier]

Abstract: The integrating factor (IF) method for numerical integration of stiff nonlinear PDEs has the disadvantage of producing large error coefficients when the linear term has large norm. We propose a generalization of the IF method, and in particular construct multistep-type methods with several orders of magnitude improved accuracy. We also consider exponential time differencing (ETD) methods, and point out connections with a particular application of the commutator-free Lie group methods. We present a new fourth order ETDRK method with improved accuracy. The methods considered are compared in several numerical examples. [Copyright &y& Elsevier]

In this paper, new Locally Linearized Runge-Kutta formulas of Dormand and Prince are introduced with the purpose of integrating large systems of initial value problems. The new formulas are obtained by replacing the Padé approximation for matrix exponential by a Krylov-Padé approximation in the embedded formulas proposed in a previous work. Unlike other high order exponential integrators, these new formulas involve the approximation of just a single phi-function times vector, which makes them particularly attractive from a computational viewpoint. With these new formulas, adaptive schemes are constructed with novelties in the approximation of phi-functions times vectors by Krylov-Padé approximations, and in the strategies for controlling the approximation errors, for estimating the dimension of the Krylov subspaces, and for the reuse of the Jacobians. In addition, numerical experiments are performed to show the potential of the new embedded formulas and adaptive codes in the integration of known physical, biophysical, and physical-chemistry models. [ABSTRACT FROM AUTHOR]