On the error structure of the implicit Euler scheme applied to stiff systems of differential equations

In this paper we investigate the structure of the global discretization error of the implicit Euler scheme applied to systems to stiff differential equations, extending earlier work on this subject (cf. [1], [9]). We restrain our considerations to the linear, self-adjoint, constant coefficient case but—in contrast to [1], [9]—we make no assumptions about the nature of the stiff spectrum; the stiff eigenvalues may be arbitrarily distributed on the negative real axis. Our main result says that the global error of the implicit Euler scheme admits an asymptotic expansion in powers of the stepsize τ which is not asymptotically correct in the conventional sense: Near the initial pointt=0 the expansion is spoiled at theO(τ²) by ‘irregular’ error components which are, however, (algebraically) damped, such that away fromt=0 the ‘pure’ asymptotic expansion reappears. We present numerical experiments confirming this result. Our considerations should be particularly helpful for a rigorous, quantitative analysis of the structure of the full (space & time) discretization error in the PDE (method of lines) context, and thus for a sound theoretical justification of extrapolation techniques for this important class of stiff problems.     

Asymptotic expansions of the global error for the implicit midpoint rule (stiff case)

In this contribution an asymptotic expansion of the global error for the box scheme (implicit midpoint rule) in a stiff case on a highly non-uniform mesh will be discussed. The details and the proofs may be found in [7].     

Multiparameter singular perturbation problems: Iterative expansions and asymptotic stability

Recursive formulae are derived which yield asymptotic expansions for the eigenvalues of multiparameter singular perturbation problems. These formulae follow readily from an exact expression for the eigenvalues which involves an implicit matrix function. The implicit function satisfies an algebraic matrix Riccati equation reminiscent of a similar equation of the single parameter theory. The results also explicate the ‘block D-stability’ criterion for asymptotic stability previously introduced by Khalil and Kokotovic.     

General linear methods for ordinary differential equations

General linear methods were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep methods. They have potential applications, especially for stiff problems. This paper discusses stiffness and emphasises the need for efficient implicit methods for the solution of stiff problems. In this context, a survey of general linear methods is presented, including recent results on methods with the inherent RK stability property.     

奇异最优控制的渐近分析

该文研究分布参数系统的奇异最优控制的收敛性和渐近分析,给出了一种可行的渐近展 开算法和误差估计,并提出一个Stiff类型的未解决问题.     

Solving stiff differential equations for simulation

Computer simulation of dynamic systems very often leads to the solution of a set of stiff ordinary differential equations. The solution of this set of equations involves the eigenvalues of its Jacobian matrix. The greater the spread in eigenvalues, the more time consuming the solutions become when existing numerical methods are employed. Extremely stiff differential equations can become a very serious problem for some systems, rendering accurate numerical solutions completely uneconomic. In this paper, we propose new techniques for solving extremely stiff systems of differential equations. These algorithms are based on a class of implicit Runge-Kutta procedure with complete error estimate. The new techniques are applied to solving mathematical models of the relaxation problem behind blast waves.     

Model equations for accuracy investigation of Runge-Kutta methods

The simplest equations are considered that simulate the behavior of various error components of Runge-Kutta methods. The expressions for the local and global errors are obtained. The minimization of these errors allows one to construct explicit and implicit methods that have an improved accuracy when solving stiff and differential-algebraic problems.     

Parallel Rosenbrock methods for chemical systems

Parallel Rosenbrock methods are developed for systems with stiff chemical reactions. Unlike classical Runge-Kutta methods, these linearly implicit schemes avoid the necessity to iterate at each time step. Parallelism across the method allows for the solution of the linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. In addition to possessing excellent stability properties, these methods are computationally efficient while preserving positivity of the solutions. Numerical results confirm these characteristics when applied to problems involving stiff chemistry, and enzyme kinetics.     

A study ofB-convergence of linearly implicit Runge-Kutta methods

This paper deals withB-convergence analysis of linearly implicit Runge-Kutta methods as applied to stiff, semilinear problems of the formy′(t)=Ty(t)+g(t,y). We analyse the discrepancy between the local and global order reduction. We show that linearly implicit Runge-Kutta methods ofB-consistency orderq have theB-convergence orderq+1 for many singularly perturbed problems with constant stiff part. Numerical examples illustrate the theoretical results.     

Asymptotic errox expansions for Defect Correction Iterates

Asymptotic error expansions are shown to exist for Iterated Defect Correction (IDeC) methods applied to differential equations. Only weak assumptions are needed to guarantee the existence of these error expansions. The results are derived in a very general way which can easily be extended to other types of operator equations.     

Combination of the boundary integral equation method and the extrapolation method

Many numerical methods including the boundary integral equation method start with division of the domain of calculation into intervals. The accuracy of their results can be improved considerably by extrapolation. To be able to apply the extrapolation method it is necessary to know the asymptotic expansion of the error.In this paper the principle of the extrapolation method and subjects important for its application are described. Above all it is shown how to determine the asymptotic expansions numerically by trial and error. In the first sections the matter is explained in a general manner to encourage users of various numerical methods—among them users of the finite element method—to try to extrapolate their results. Then the investigations are exemplified in detail by the boundary integral equation method. The accuracy of approximate solutions of integral equations for plane elastostatic problems with prescribed boundary tractions and displacements is improved by extrapolation. Particular attention is paid to boundary tractions and displacements with discontinuous derivatives.To induce also practically orientated readers without specialized mathematical knowledge to think about applying the extrapolation method the basic topics are represented in an extensive manner and illustrated by simple examples. (For a survey of this paper see end of Section 1.)     

Initial value problems: numerical methods and mathematics

A number of questions and results concerning Runge-Kutta and general linear methods are surveyed. These include order conditions and order bounds for Runge-Kutta methods, the A-stability of implicit Runge-Kutta methods based on Gaussian quadrature and transformation methods of implementation which lead to singly-implicit methods. The sections dealing with general linear methods include a discussion of the order conditions and an algebraic structure for carrying out order analyses as well as an introduction to a special function associated with parallel methods for stiff problems.     

Automatic h-Scaling for the Efficient Time Integration of Stiff Mechanical Systems

When solving stiff mechanical systems, implicit time integrators overestimate the error and tend to use small stepsizes due to the order reduction phenomenon. This article introduces an algorithm for the detection of stiff components that allows automatic scaling of the stepsize h in the error estimation. It is based on an investigation of the local error and applies existing embedded formulas of implicit Runge–Kutta methods. Thus, implementation and calculation effort are low. Three examples are integrated with the code RADAU5, a linear DAE in two different formulations and a slider crank mechanism.     

Stochastic projective methods for simulating stiff chemical reacting systems

In this paper, stochastic projective methods are proposed to improve the stability and efficiency in simulating stiff chemical reacting systems. The efficiency of existing explicit tau-leaping methods can often severely be limited by the stiffness in the system, forcing the use of small time steps to maintain stability. The methods presented in this paper, namely stochastic projective (SP) and telescopic stochastic projective (TSP) method, can be considered as more general stochastic versions of the recently developed stable projective numerical integration methods for deterministic ordinary differential equations. SP and TSP method are developed by fully re-interpreting and extending the key projective integration steps in the deterministic regime under a stochastic context. These new stochastic methods not only automatically reduce to the original deterministic stable methods when applied to simulating ordinary differential equations, but also carry the enhanced stability property over to the stochastic regime. In some sense, the proposed methods are stochastic generalizations to their deterministic counterparts. As such, SP and TSP method can adopt a much larger effective time step than is allowed for explicit tau-leaping, leading to noticeable runtime speedup. The explicit nature of the proposed stochastic simulation methods relaxes the need for solving any coupled nonlinear systems of equations at each leaping step, making them more efficient than the implicit tau-leaping method with similar stability characteristics. The efficiency benefits of SP and TSP method over the implicit tau-leaping is expected to grow even more significantly for large complex stiff chemical systems involving hundreds of active species and beyond.     

Stabilized starting algorithms for collocation Runge-Kutta methods

In this paper, we propose a technique to stabilize some starting algorithms often used in the Newton-type iterations appearing when collocation Runge-Kutta methods are applied to solve stiff initial value problems. By following the ideas given in [1], we analyze the order (classical and stiff) of the new starting algorithms and pay special attention to their error amplifying functions. From the computational point of view, the new algorithms require the solution of an additional linear system per integration step, but as shown in the numerical experiments, this extra cost is compensated in most of the problems by their better stability properties.     

Optimal asymptotic identification under bounded disturbances

The intrinsic limitation of worst-case identification of linear time-invariant systems using data corrupted by bounded disturbances, when the unknown plant is known to belong to a given model set, is studied. This is done by analyzing the optimal worst-case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm. It is shown that under some topological conditions on the model set, there is an identification algorithm which is asymptotically optimal for any input, and the optimal asymptotic error is characterized as a function of the inputs. These results, which hold for any error metric and disturbance norm, are applied to three specific identification problems: identification of stable systems in the l₁ norm, identification of stable rational systems in the H_∞ norm and identification of unstable rational systems in the gap metric. For each of these problems, the general characterization of optimal asymptotic error is used to find near-optimal inputs to minimize the error     

Richardson Extrapolation of Iterated Discrete Galerkin Method for Eigenvalue Problem of a Two Dimensional Compact Integral Operator

We consider approximation of eigenelements of a two-dimensional compact integral operator with a smooth kernel by discrete Galerkin and iterated discrete Galerkin methods. By choosing numerical quadrature appropriately, we obtain superconvergence rates for eigenvalues and iterated eigenvectors, and for gap between the spectral subspaces. We propose an asymptotic error expansions of the iterated discrete Galerkin method and asymptotic error expansion of approximate eigenvalues. We then apply Richardson extrapolation to obtain improved error bounds for the eigenvalues. Numerical examples are presented to illustrate theoretical estimate.     

Partitioned and Implicit–Explicit General Linear Methods for Ordinary Differential Equations

Implicit–explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiff components. IMEX Runge–Kutta methods and IMEX linear multistep methods have been studied in the literature. In this paper we study new implicit–explicit methods of general linear type. We develop an order conditions theory for high stage order partitioned general linear methods (GLMs) that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order up to three. Numerical results confirm the theoretical findings.     

Efficient implementation of second-order implicit Runge-Kutta methods

Implementation schemes for second-order implicit Runge-Kutta methods are considered. The schemes allow one to reduce computational costs when solving stiff problems with low accuracy. The results of the comparison with implicit MATLAB solvers are presented.