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We note a fact that stiff systems or differential equations that have highly oscillatory solutions cannot be solved efficiently using conventional methods. In this paper, we study two new classes of exponential Runge–Kutta (ERK) integrators for efficiently solving stiff systems or highly oscillatory problems. We first present a novel class of explicit modified version of exponential Runge–Kutta (MVERK) methods based on the order conditions. Furthermore, we consider a class of explicit simplified version of exponential Runge–Kutta (SVERK) methods. Numerical results demonstrate the high efficiency of the explicit MVERK integrators and SVERK methods derived in this paper compared with the well-known explicit ERK integrators for stiff systems or highly oscillatory problems in the literature. [ABSTRACT FROM AUTHOR]

Unified constitutive equations have been developed to model the behaviour of metallic materials under various processing conditions. These constitutive equations usually take the form of a set of ordinary differential equations (ODEs), which must be solved thousands of times in a finite element (FE) process simulation. Thus, an efficient and reliable numerical integration method for large systems is crucial for solving this problem. However, in many constitutive equations, numerical stiffness is often present. This means that the stability requirements, rather than the accuracy, constrain the step size. Therefore, certain numerical methods become unsuitable when the required step size becomes unacceptably small. In this study, a series of mathematical analyses was performed to investigate the difficulties in the numerical integration of three sets of unified viscoplastic/creep constitutive equations. Based on an analysis of the current stiffness assessment methods, a novel index was introduced, that enables an accurate assessment of the stiffness of the ODE-type unified constitutive equations. A computational study was also conducted to benchmark several promising implicit numerical integration methods for viscoplastic/creep constitutive equations. This study can assist researchers in metal forming and other fields in choosing appropriate numerical methods when dealing with stiff ODEs.

In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are A-stable for order p, (p = 4 , 5 , 6 , 7) and A (α )-stable for order p, (p = 8 , 9 , ... , 14). Compared to the existing BDF based method, i.e. class 2 + 1 , hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF. [ABSTRACT FROM AUTHOR]

This paper presents the innovative Taylor wavelet collocation method (TWCM) for the stiff systems arising in chemical reactions. In this technique, first, we generated the functional matrix of integration (FMI) for the Taylor wavelets. Using this FMI, the Taylor wavelet collocation method is proposed to obtain the numerical approximation of stiff systems in the form of a system of ordinary differential equations (SODEs). This method converts the SODEs into a set of algebraic equations, which can be solved by the Newton–Raphson method. To demonstrate the simplicity and effectiveness of the presented approach, numerical results are obtained. Graphs and tables illustrate the created strategy's effectiveness and consistency. Illustrative examples are examined to demonstrate the performance and effectiveness of the developed approximation technique, and a comparison is made with the current results. Results reveal that the newly selected strategy is superior to previous approaches regarding precision and effectiveness in the literature. Most semi-analytical and numerical methods work based on controlling parameters, but this technique is free from controlling parameters. Also, it is easy to implement and consumes less time to handle the system. The suggested wavelet-based numerical method is computationally appealing, successful, trustworthy, and resilient. All computations have been made using the Mathematica 11.3 software. The convergence of this strategy is explained using theorems. [ABSTRACT FROM AUTHOR]

Unified constitutive equations have been developed to model the behaviour of metallic materials under various processing conditions. These constitutive equations usually take the form of a set of ordinary differential equations (ODEs), which must be solved thousands of times in a finite element (FE) process simulation. Thus, an efficient and reliable numerical integration method for large systems is crucial for solving this problem. However, in many constitutive equations, numerical stiffness is often present. This means that the stability requirements, rather than the accuracy, constrain the step size. Therefore, certain numerical methods become unsuitable when the required step size becomes unacceptably small. In this study, a series of mathematical analyses was performed to investigate the difficulties in the numerical integration of three sets of unified viscoplastic/creep constitutive equations. Based on an analysis of the current stiffness assessment methods, a novel index was introduced, that enables an accurate assessment of the stiffness of the ODE-type unified constitutive equations. A computational study was also conducted to benchmark several promising implicit numerical integration methods for viscoplastic/creep constitutive equations. This study can assist researchers in metal forming and other fields in choosing appropriate numerical methods when dealing with stiff ODEs. [ABSTRACT FROM AUTHOR]

Abstract Implicit block approaches are used by a number of numerical analyzers to model mild, medium, and hard differential systems. Their excellent stability characteristics, self-starting nature, quick convergence, and large decrease in computing cost all contribute to their widespread application. With these numerical benefits in mind, a new one-step implicit block method with three intrastep grid points has been created. The major term of the local truncation error is minimized to determine which of these points is optimal. The reformulation of the suggested technique leads to a significant decrease in computing cost while maintaining the same consistency, zero-stability, A $\mathcal{A}$ -stability, and convergence. Several sorts of error are calculated, together with CPU time and efficiency plot, to determine which is superior. Differential models from the fields of heat transfer, population dynamics, and chemical engineering show that the suggested method does a better job than some of the current hybrid block and implicit Radau methods with similar properties.

The Linear Implicit Quantized State System (LIQSS) method has been evaluated for suitability in modeling and simulation of long duration mission profiles of Naval power systems which are typically characterized by stiff, non-linear, differential algebraic equations. A reference electromechanical system consists of an electric machine connected to a torque source on the shaft end and to an electric grid at its electrical terminals. The system is highly non-linear and has widely varying rate constants; at a typical steady state operating point, the electrical and electromechanical time constants differ by three orders of magnitude—being 3.2 ms and 2.7 s respectively. Two important characteristics of the simulation—accuracy and computational intensity—both depend on quantization size of the system state variables. At a quantization size of about 1% of a variable's maximum value, results from the LIQSS1 method differed by less than 1% from results computed by well-known continuous-system state-space methods. The computational efficiency of the LIQSS1 method increased logarithmically with increasing quantization size, without significant loss of accuracy, up to some particular quantization size, beyond which the error increased rapidly. For the particular system under study, a "sweet spot" was found at a particular quantum size that yielded both high computational efficiency and good accuracy. [ABSTRACT FROM AUTHOR]

The size and complexity of multi-scale problems such as those arising in chemical kinetics mechanisms has stimulated the search for methods that reduce the number of species and chemical reactions but retain a desired degree of accuracy. The time-scale characterisation of the multi-scale problem can be carried out on the basis of local information such as the Jacobian matrix of the model problem and its related eigen-system evaluated at one point P of the system trajectory. While the original problem is usually described by ordinary differential equations (ODEs), the reduced order model is described by a reduced number of ODEs and a number of algebraic equations (AEs), that might express one or more physical conservation laws (mass, momentum, energy), or the fact that the long-term dynamics evolves within a so-called Slow Invariant Manifold (SIM). To fully exploit the benefits offered by a reduced order model, it is required that the time scale characterisation of the n-dimensional reduced order model returns an answer consistent and coherent with the time-scale characterisation of the N-dimensional original model. This manuscript discusses a procedure for obtaining the time-scale characterisation of the reduced order model in a manner that is consistent with that of the original problem. While a standard time scale characterisation of the (original) N-dimensional original model can be carried out by evaluating the eigen-system of the ( N × N) Jacobian matrix of the vector field that defines the system dynamics, the time-scale characterisation of the n-dimensional reduced order model (with n

Implicit block approaches are used by a number of numerical analyzers to model mild, medium, and hard differential systems. Their excellent stability characteristics, self-starting nature, quick convergence, and large decrease in computing cost all contribute to their widespread application. With these numerical benefits in mind, a new one-step implicit block method with three intrastep grid points has been created. The major term of the local truncation error is minimized to determine which of these points is optimal. The reformulation of the suggested technique leads to a significant decrease in computing cost while maintaining the same consistency, zero-stability, A -stability, and convergence. Several sorts of error are calculated, together with CPU time and efficiency plot, to determine which is superior. Differential models from the fields of heat transfer, population dynamics, and chemical engineering show that the suggested method does a better job than some of the current hybrid block and implicit Radau methods with similar properties. [ABSTRACT FROM AUTHOR]

In this paper, a new family of second order two-step Runge–Kutta–Chebyshev methods is presented. These methods are a generalization of the one-step stabilized methods and have better computational properties compared to them. A new code TSRKC2 is developed and compared to existing solvers at sufficiently large set of examples. [Display omitted] [ABSTRACT FROM AUTHOR]

In the construction of efficient numerical methods for the stiff initial value problems, some second derivative multistep methods have been introduced equipping with super future point technique. In this paper, we are going to introduce a formula for the stability functions of a class of such methods. This group of methods encompasses SDBDF methods and their extensions with advanced step-point feature. This general formula, instead of obtaining the distinct stability functions for each of methods, will facilitate stability analysis of the methods. [ABSTRACT FROM AUTHOR]

Abstract A hybrid second derivative three-step method of order 7 is proposed for solving first order stiff differential equations. The complementary and main methods are generated from a single continuous scheme through interpolation and collocation procedures. The continuous scheme makes it easy to interpolate at off-grid and grid points. The consistency, stability, and convergence properties of the block formula are presented. The hybrid second derivative block backward differentiation formula is concurrently applied to the first order stiff systems to generate the numerical solution that do not coincide in time over a given interval. The numerical results show that the new method compares favorably with some known methods in the literature.

Solving stiff, singular, and singularly perturbed initial value problems (IVPs) has always been challenging for researchers working in different fields of science and engineering. In this research work, an attempt is made to devise a family of nonlinear methods among which second‐ to fourth‐order methods are not only 풜− stable but 풜− acceptable as well under order stars' conditions. These features make them more suitable for solving stiff and singular systems in ordinary differential equations. Methods with remaining orders are either zero‐ or conditionally stable. The theoretical analysis contains local truncation error, consistency, and order of accuracy of the proposed nonlinear methods. Furthermore, both fixed and variable stepsize approaches are introduced wherein the latter improves the performance of the devised methods. The applicability of the methods for solving the system of IVPs is also described. When used to solve problems from physical and real‐life applications, including nonlinear logistic growth and stiff model for flame propagation, the proposed methods are found to have good results. [ABSTRACT FROM AUTHOR]

The present study proposes a new explicit nonlinear scheme that solves stiff and nonlinear initial value problems in ordinary differential equations. One of the promising features of this scheme is its fourth-order convergence with strong stability having an unbounded region. A modern approach for relative stability growth analysis is also presented under order stars conditions. The scheme is also good in dealing with singular and stiff type of models. Comparing numerical experiments using various errors, including maximum absolute global error over the integration interval, absolute error at the endpoint, average error, norm of errors, and the CPU times (seconds), shows better performance. An adaptive step-size approach seems to increase the performance of the proposed scheme. The numerical simulations assure us of L-stability, consistency, order, and rapid convergence. [ABSTRACT FROM AUTHOR]

In this work, a novel class of collocation methods for numerical integration of ODEs is presented. Methods are derived from the weighted integral form of ODEs by assuming that a polynomial function at individual time increment approximates the solution of the ODE. A distinct feature of the approach, which we demonstrated in this work, is that it allows the increase of accuracy of a method while retaining the number of method coefficients. This is achieved by applying different quadrature rule to the approximation function and the ODE, resulting in different behaviour of a method. Quadrature rules that we examined in this work are the Gauss–Legendre and Lobatto quadrature where several other quadrature rules could further be explored. The approach has also the potential for enhancing the accuracy of the established Runge–Kutta-type methods. We formulated the methods in the form of Butcher tables for convenient implementation. The performance of the new methods is investigated on some well-known stiff, oscillatory and non-linear ODEs from the literature. [ABSTRACT FROM AUTHOR]

An approach to mathematically model the nucleation and growth of soot particles during the diffusion combustion of hydrocarbon fuel is presented. The chain of transformations of hydrocarbon fuel is modeled using a Markov process with a finite number of states, which is described by a stiff system of Kolmogorov's ordinary differential equations (ODEs). As a result of numerical modeling, the set of functions describing the change in concentrations of various soot fractions in a laminar diffusion flame of toluene depending on time is obtained. Based on the results of the numerical solution of the ODE system, discrete particle size distributions (relative diameters) are constructed for different times. Continuous Weibull distributions approximating discrete distributions are constructed using the least squares method. The results obtained are in qualitative agreement with the experimental data. [ABSTRACT FROM AUTHOR]

A modified Newton scheme is typically used to solve large sets of non-linear equations arising in the implementation of implicit Runge-Kutta methods. As an alternative to this scheme, iteration schemes, which sacrifice super linear convergence for reduced linear algebra costs, have been proposed. A more general linear iterative scheme of this type proposed by Cooper and Butcher in 1983 for implicit Runge-Kutta methods, and he has applied the successive over relaxation technique to improve the convergence rate. In this paper, we establish the convergence result of this scheme by proving some theoretical results suitable for stiff problems. Also these convergence results are verified by two and three stage Gauss method and Radue IIA method. [ABSTRACT FROM AUTHOR]

A hybrid second derivative three-step method of order 7 is proposed for solving first order stiff differential equations. The complementary and main methods are generated from a single continuous scheme through interpolation and collocation procedures. The continuous scheme makes it easy to interpolate at off-grid and grid points. The consistency, stability, and convergence properties of the block formula are presented. The hybrid second derivative block backward differentiation formula is concurrently applied to the first order stiff systems to generate the numerical solution that do not coincide in time over a given interval. The numerical results show that the new method compares favorably with some known methods in the literature. [ABSTRACT FROM AUTHOR]

This paper introduces a new class of exponential integrators based on spectral deferred correction. These new methods are simple to implement at any order of accuracy and can be used to efficiently solve initial value problems when high precision is desired. We begin by deriving exponential spectral deferred correction (ESDC) methods for solving both partitioned and unpartitioned initial value problems. We then analyze the linear stability properties of these new integrators and show that they are comparable to those of existing semi-implicit spectral deferred correction schemes. Finally, we present five numerical experiments to demonstrate the improved efficiency of our new exponential integrator compared to semi-implicit spectral deferred correction schemes and existing fourth-order exponential Runge--Kutta methods. [ABSTRACT FROM AUTHOR]

The stochastic Galerkin approach is a widely used method for quantifying uncertainty in dynamical systems. A polynomial chaos expansion of physical variables is used to represent uncertainty. Thus the evolution equations for the dynamics of physical quantities are replaced by evolution equations for the polynomial chaos coefficients. The dimension of the stochastic Galerkin system is m times larger than the dimension of the original physical system, where m is the number of polynomial chaos coefficients. The cost of implicit time integration methods, required for stiff dynamical systems, increases considerably, for example, by a factor m 3 when direct linear solvers are used. This work studies efficient implicit methods for the time integration of stiff stochastic Galerkin systems consisting of ordinary differential equations. Linearly-implicit Rosenbrock–Wanner schemes are considered. A block-diagonal approximation of the Jacobian of the stochastic Galerkin system is employed, where the diagonal blocks are the Jacobian of the physical system, evaluated at the mean solution value. We perform a theoretical study of numerical stability for this approximation, using a stochastic extension of the Dahlquist test equation, and propose a concept of linear stochastic Galerkin stability. A rigorous analysis is done for methods of order one and two. A third-order scheme is designed to investigate the stability properties by numerical experiments. Numerical results confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

Stabilized methods for the numerical solution of ODEs, also called Chebyshev methods, are explicit methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. In this paper we present explicit two-step Runge–Kutta–Chebyshev methods of order two, which have more than 2.3 times larger stability intervals than the analogous one-step methods. Explicit formulae for stability intervals are derived, as well as an effective recurrent scheme for calculation of methods' coefficients for arbitrary number of stages. Our numerical experiments confirm the accuracy and stability properties of the proposed methods and show that at least in the case of constant time steps they can compete with the well-known ROCK2 method. [Display omitted] [ABSTRACT FROM AUTHOR]

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order.

We propose the S-leaping algorithm for the acceleration of Gillespie's stochastic simulation algorithm that combines the advantages of the two main accelerated methods; the τ -leaping and R-leaping algorithms. These algorithms are known to be efficient under different conditions; the τ -leaping is efficient for non-stiff systems or systems with partial equilibrium, while the R-leaping performs better in stiff system thanks to an efficient sampling procedure. However, even a small change in a system's set up can critically affect the nature of the simulated system and thus reduce the efficiency of an accelerated algorithm. The proposed algorithm combines the efficient time step selection from the τ -leaping with the effective sampling procedure from the R-leaping algorithm. The S-leaping is shown to maintain its efficiency under different conditions and in the case of large and stiff systems or systems with fast dynamics, the S-leaping outperforms both methods. We demonstrate the performance and the accuracy of the S-leaping in comparison with the τ -leaping and R-leaping on a number of benchmark systems involving biological reaction networks. [ABSTRACT FROM AUTHOR]

An implicit scheme with splitting with respect to physical processes is proposed for a stiff system of two-dimensional Euler gas dynamics equations with chemical source terms. For the first time, convection is computed using a bicompact scheme that is fourth-order accurate in space and third-order accurate in time. This high-order bicompact scheme is L-stable in time. It employs a conservative limiting method and Cartesian meshes with solution-based adaptive mesh refinement. The chemical reactions are computed using an L-stable second-order Runge–Kutta scheme. The developed scheme is successfully tested as applied to several problems concerning detonation wave propagation in a two-species ideal gas with a single combustion reaction. The advantages of bicompact schemes over the popular MUSCL and WENO5 schemes as applied to shock-capturing computations of detonation waves are discussed. [ABSTRACT FROM AUTHOR]

In this article we propose a modification to Linearly Implicit Quantized State System Methods (LIQSS), a family of methods for solving stiff Ordinary Differential Equations (ODEs) that replace the classic time discretization by the quantization of the state variables. LIQSS methods were designed to efficiently simulate stiff systems, but they only work when the system has a particular structure. The proposed modification overcomes this limitation, allowing the algorithms to efficiently simulate stiff systems with more general structures. Besides describing the new methods and their algorithmic descriptions, the article analyzes the algorithms performance in the simulation of some complex systems. [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]

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

The article provides an overview of cases of stiffness, leading to inadequate results of mathematical models of mechanics. An attempt is made to generalize look at this phenomenon for body dynamics, and for the continuum mechanics. Proposed wording of display of stiffness properties of the mechanical systems, the primary cause of unstable solutions and makes recommendations for the development of sustainable calculation metods.

The new set of absolutely stable difference schemes for a numerical solution of ODEs stiff systems (1) is submitted: ( ) ( ), 0, (0) 0 d u t f u t u u dt    . (1) The main feature of the set is the multi-implicit finite differences with the second derivatives of the desired solution. The expanded three-parameter (, , ) set of 2ISD-schemes (2)-(3) is studied in more details in this paper. 2 1 1 1 0 2 2 2 2 0 ( ф ) ф ( ф ) 2ф n n i i n i n i i n n i i n i n i i a E b J f a E b J f                       v v v v (2)     101 128 11 240 240 240 56 128 56 240 240 240 13 40 3 240 240 240 8 8 240 240 3б 2в 4в 3б 2в , 3г 3г б в 4б б в г 4г г ki ki a b                                     (3) At arbitrary (, , ) parameters last difference equation in system (2) has 5th order of accuracy. We found that the set of absolutely stable 2ISD-schemes includes two families: the set of the L-stable schemes and the set of the schemes of heightened accuracy for linear problems. For example: at 1 б  168 , в  0, г  0 we have A-stable scheme with 8th order of approximation, at 53 1 6 б   5880 , в  148 , г  315 we have L1-stable scheme with 7th order of approximation, at 23 1 14 б   360 , в  60 , г  315 we have L2-stable scheme with 6th order of approximation. The testing of this difference schemes on linear and nonlinear problems with a different stiff power is conducted. The errors of a numerical solution as functions of integration step size are computed in numerical experiments. These results demonstrate high quality of stability and accuracy of the suggested 2ISD-schemes

A new class of hybrid BDF-like methods is presented for solving systems of ordinary differential equations (ODEs) by using the second derivative of the solution in the stage equation of class 2 + 1hybrid BDF-like methods to improve the order and stability regions of these methods. An off-step point, together with two step points, has been used in the first derivative of the solution, and the stability domains of the new methods have been obtained by showing that these methods are A-stable for order p, p = 3,4,5,6,7and A(α)-stable for order p, 8 ≤ p ≤ 14. The numerical results are also given for four test problems by using variable and fixed step-size implementations. [ABSTRACT FROM AUTHOR]

In many applications, large systems of ordinary differential equations with both stiff and nonstiff parts have to be solved numerically. Implicit-explicit (IMEX) methods are useful for efficiently solving these problems. In this paper, we construct IMEX second-derivative BDF methods with considerable stability properties. To show the efficiency of the introduced technique, numerical comparisons are given by solving some problems. [ABSTRACT FROM AUTHOR]

Long time integration of large stiff systems of initial value problems, arising from chemical reactions, demands efficient methods with good accuracy and extensive absolute stability region. In this paper, we apply second derivative general linear methods to solve some stiff chemical problems such as chemical Akzo Nobel problem, HIRES problem and OREGO problem.

In this paper we consider the development of Implicit-Explicit (IMEX) Runge{Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior, which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge{Kutta methods for hyperbolic systems with relaxation lose their efficiency, and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable of capturing the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Implicit–Explicit (IMEX) schemes are a powerful tool in the development of numerical methods for hyperbolic systems with stiff sources. Here we focus our attention on the asymptotic properties of such schemes, like the preservation of steady-states (well-balanced property) and the behavior in presence of small space–time scales (asymptotic preservation property). We analyze conditions under which the standard additive approach based on taking the fluxes explicitly and the sources implicitly yields a well-balanced behavior. In addition, we consider a partitioned strategy which possesses better well-balanced properties. The behavior of the additive and partitioned approaches under classical scaling limits is then studied in the context of asymptotic-preserving schemes. Additional order conditions that guarantee the correct behavior of the schemes in the Navier–Stokes regime are derived. Several examples illustrate these asymptotic behaviors and the performance of the new methods. [ABSTRACT FROM AUTHOR]

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr-Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641-666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit-explicit (IMEX) Runge-Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr-Debye model. The new IMEX Runge-Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge-Kutta method and have second-order accuracy. The numerical results validate our analysis. [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]

Considering the methods with future points technique from second derivative general linear methods (SGLMs) point of view, makes it possible to improve their stability properties. In this paper, we extend the stability regions of a modified version of E2BD formulas to optimal one and show its effectiveness by numerical verifications. Also, implementation issues, with numerical experiments, of these methods are investigated in a variable step-size mode. [ABSTRACT FROM AUTHOR]