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High order methods are often desired for the evolution of ordinary differential equations, in particular those arising from the semidiscretization of partial differential equations. In prior work w...

We present a numerical method for convergence acceleration for multifidelity models of parameterized ordinary differential equations. The hierarchy of models is defined as trajectories computed usi...

As a sequel to our previous work [Q. Zhang, SIAM J. Numer. Anal., 51 (2013), pp. 2822--2850], [Q. Zhang and A. Fogelson, SIAM J. Sci. Comput., 36 (2014), pp. A2369--A2400], this paper presents MARS, a generic framework for analyzing interface tracking (IT) methods via mapping and adjusting regular semialgebraic sets. Our mathematical model for moving material regions is the metric space of bounded regular semianalytic sets, equipped with Boolean algebras and advected by homeomorphic flow maps of a nonautonomous ordinary differential equation. By examining the actions of semidiscrete and discrete flow maps upon this metric space, we pinpoint in Lemma 3.9 a fundamental difficulty in achieving an IT accuracy higher than the second order. We then propose a generic IT method by concatenating three unitary operations on the modeling space, bound its overall IT error by the sum of intuitively defined error terms, and further estimate the individual errors in terms of the time step size and a Lagrangian length sc...

In this paper, the three-dimensional compressible Navier--Stokes equations are considered on a periodic domain. We propose a semidiscrete numerical scheme and derive a priori bounds that ensure that the resulting system of ordinary differential equations (ODEs) is solvable for any $h>0$. An a posteriori observation that density remains uniformly bounded away from 0 will establish that a subsequence of the numerical solutions converges to a specific form of weak solution of the compressible Navier--Stokes equations.

We present improved convergence results for the Boosting Algorithm (BA), and demonstrate that an existing formulation of the Heterogeneous Multiscale Methods (HMM) is accurate to first order only in the macro time step, regardless of the order of the numerical solvers employed. These results are obtained by considering the BA and two other formulations of HMM as special cases of a general formulation of HMM applied to dissipative stiff ordinary differential equations., Comment: Accepted for publication in SINUM

We present optimal error estimates for spectral Petrov--Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov--Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates.

In this paper, we develop a family of second-order semi-implicit time integration methods for systems of ordinary differential equations (ODEs) with stiff damping term. The important feature of the new methods resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only. The new semi-implicit methods are based on the modification of explicit strong stability preserving Runge--Kutta (SSP-RK) methods and are proven to have a formal second order of accuracy, $A(\alpha)$-stability, and stiff decay. We illustrate the performance of the proposed SSP-RK based semi-implicit methods on both a scalar ODE example and a system of ODEs arising from the semi-discretization of the shallow water equations with stiff friction term. The obtained numerical results clearly demonstrate that the ability of the introduced ODE solver to exactly preserve equilibria plays ...

In practical computation with Runge-Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.

Volume-of-fluid (VOF) methods are widely-used for the interface tracking problem, yet rigorous analyses of them are rare. This paper presents such an analysis for incompressible flows by combining the theories of ordinary differential equations, differential geometry, and Boolean algebra. Based on the concept of donating region (DR) [Q. Zhang, SIAM Rev., 55 (2013), pp. 443--461], the author classifies the fluxing particles of a fixed control volume into four categories and derives three analytical solutions for the advection equation of the color function. One edgewise solution provides a unified view of DR-based advection algorithms for VOF methods while another cellwise solution serves as the theoretical foundation of the recent polygonal area mapping method. The well-known second-order convergence rates of streamline-based VOF advection algorithms are proved rigorously. Potential deterioration of this second-order convergence is also discussed for several subtle issues such as exact mass conservation a...

Peer two-step methods have been successfully applied to initial value problems for stiff and nonstiff ordinary differential equations (ODEs) both on parallel and sequential computers. Their essential property is the use of several stages per time step with the same accuracy. As a new application area these methods are now used for parameter-dependent ODEs where the peer stages approximate the solution also at different places in the parameter space. The main interest here is sensitivity data through an approximation of solution derivatives in different parameter directions. Basic stability and convergence properties are discussed and peer methods of order 2 and 3 in the time stepsize are constructed. The computed sensitivity matrix is used in approximate Newton and Gauss--Newton methods for shooting in boundary value problems, where initial values and/or ODE parameters are searched for, and in parameter identification from partial information on trajectories.

The phase flow method, originally introduced in [L. X. Ying and E. J. Candes, J. Comput. Phys., 220 (2006), pp. 184-215], can efficiently solve autonomous ordinary differential equations. In [S. Jin, H. Wu, and Z. Y. Huang, SIAM J. Sci. Comput., 31 (2008), pp. 1303-1321], the method was generalized to solve Hamiltonian system where the Hamiltonian function was discontinuous. However, both of these methods require a phase flow map constructed on an invariant manifold. This can increase computational cost when the invariant domain is big or unbounded. Following the idea of [S. Jin, H. Wu, and Z. Y. Huang, SIAM J. Sci. Comput., 31 (2008), pp. 1303-1321], we propose a hybrid phase flow method for solving the Liouville equation in a bounded domain, which is smaller than the invariant manifold of a phase flow map. By using some proper boundary conditions, this method can help solve the problem where the invariant manifold of a phase flow map determined by the Liouville equation is unbounded. We verify numerical accuracy and efficiency by several examples of the semiclassical limit of the Schrodinger equation. Analysis of numerical stability and convergence is given for the semiclassical limit equation with inflow boundary condition.

We study in this paper a new class of waveform relaxation algorithms for large systems of ordinary differential equations arising from discretizations of partial differential equations of advection reaction diffusion type. We show that the transmission conditions between the subsystems have a tremendous influence on the convergence speed of the waveform relaxation algorithms, and we identify transmission conditions with optimal performance. Since these optimal transmission conditions are expensive to use, we introduce a class of local transmission conditions of Robin type, which approximate the optimal ones and can be used at the same cost as the classical transmission conditions. We determine the transmission conditions in this class with the best performance of the associated waveform relaxation algorithm. We show that the new algorithm is well posed and converges much faster than the classical one. We illustrate our analysis with numerical experiments.

Space discretization of some time‐dependent partial differential equations gives rise to ordinary differential equations containing additive terms with different stiffness properties. In these situ...

We consider the behavior of certain adaptive timestepping methods, based upon embedded explicit Runge--Kutta pairs, when applied to dissipative ODEs. It has been observed numerically that the standard local error controls can impart desirable stability properties, but this has been rigorously verified only for very special, low-order, Runge--Kutta pairs. The rooted-tree expansion of a certain quadratic form, central to the stability theory of Runge--Kutta methods, is derived. This, together with key assumptions on the sequence of accepted time-steps and the local error estimate, provides a general explanation for the observed stability of such algorithms on dissipative problems. Under these assumptions, which are expected to hold for "typical" numerical trajectories, two different results are proved. First, for a large class of embedded Runge--Kutta pairs of order $(1,2)$, controlled on an error-per-unit-step basis, all such numerical trajectories will eventually enter a particular bounded set. This occurs for sufficiently small tolerances independent of the initial conditions. Second, for pairs of arbitrary orders $(p-1,p)$, operating under either error-per-step or error-per-unit-step control, similar results are obtained when an additional structural assumption (that should be valid for many cases of interest) is imposed on the dissipative vector field. Numerical results support both the analysis and the assumptions made.

We provide a theoretical analysis of the processing technique for the numerical integration of ODEs. We get the effective order conditions for processed methods in a general setting so that the results obtained can be applied to different types of numerical integrators. We also propose a procedure to approximate the postprocessor such that its evaluation is virtually cost-free. The analysis is illustrated for a particular class of composition methods.

An exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow. We present a general-purpose method for proving the existence of finite-time shadows of numerical ODE integrations of arbitrary dimension in which some measure of hyperbolicity is present and there are either 0 or 1 expanding modes, or 0 or 1 contracting modes. Much of the rigor is provided automatically by interval arithmetic and validated ODE integration software that is freely available. The method is a generalization of a previously published containment process that was applicable only to two-dimensional maps. We extend it to handle maps of arbitrary dimension with the above restrictions, and finally to ODEs. The method involves building n-cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times. The proof consists of two steps: first, the rigorous computational verification of a simple geometric property, which we call the inductive containment property, and second, a simple geometric argument showing that this property implies the existence of a shadow. The computational step is almost entirely automated and easily adaptable to any ODE problem. The method allows for the rescaling of time, which is a necessary ingredient for successfully shadowing ODEs. Finally, the method is local, in the sense that it builds the shadow inductively, requiring information only from the most recent integration step, rather than more global information typical of several other methods. The method produces shadows of comparable length and distance to all currently published results. Finally, we conjecture that the inductive containment property implies the existence of a shadow without restriction on the number of expanding and contracting modes, although proof currently eludes us.

We analyze Runge--Kutta discretizations applied to index 2 differential algebraic equations (DAEs). The asymptotic features of the numerical and the exact solutions are compared. It is shown that Runge--Kutta methods satisfying the first order constraint condition of the DAE correctly reproduce the geometric properties of the continuous system. The proof combines embedding techniques of index 2 DAEs and ordinary differential equations (ODEs) with some invariant manifolds results of Nipp and Stoffer [Attractive Invariant Manifolds for Maps, SAM Research Report 92-11, ETH, Zurich, Switzerland, 1992]. The results support the favorable behavior of these Runge--Kutta methods applied to index 2 DAEs for $t \ge 0$.

For models described by ordinary differential equation boundary value problems (ODE BVPs), we derive adjoint equations for sensitivity analysis, giving explicit forms for the boundary conditions of the adjoint boundary value problem. The solutions of the adjoint equations are used to efficiently compute gradients of both integral-form and pointwise constraints. Existence and stability results are given for the adjoint system and its numerical solution. The use of the method is demonstrated for a simple example, where it is seen that the method is particularly advantageous for problems with more than a few parameters.

A new method for constructing square-conservative schemes for a class of evolution equations using Lie-group methods is presented. The basic idea is as follows. First, we discretize the space variable appropriately so that the resulting semidiscrete system of equations can be cast into a system of ordinary differential equations evolving on a sphere. Second, we apply Lie-group methods to the semidiscrete system, and then square-conservative schemes can be constructed since the obtained numerical solution evolves on the same sphere. Both exponential and Cayley coordinates are used. Numerical experiments are also reported.

We establish a rate of convergence for a semidiscrete operator splitting method applied to Hamilton--Jacobi equations with source terms. The method is based on sequentially solving a Hamilton--Jacobi equation and an ordinary differential equation. The Hamilton--Jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit Euler method. We prove that the $L^{\infty}$ error associated with the operator splitting method is bounded by $\mathcal{O}(\Delta t)$, where $\Delta t$ is the splitting (or time) step. This error bound is an improvement over the existing $\mathcal{O}(\sqrt{\Delta t})$ bound due to Souganidis [Nonlinear Anal., 9 (1985), pp. 217--257]. In the one-dimensional case, we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous Hamilton--Jacobi equations. It is proved that this fully discrete splitting method possesses a linear convergence rate. Moreover, numerical results are presented to illustrate the theoretical convergence results.

We investigate the convergence properties of the iterated defect correction (IDeC) method based on the implicit Euler rule for the solution of singular initial value problems with a singularity of the first kind. We show that the method retains its classical order of convergence, which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure.

We consider possibly stiff and implicit systems of ordinary differential equations (ODEs). The major difficulty and computational bottleneck in the implementation of fully implicit Runge--Kutta (IRK) methods resides in the numerical solution of the resulting systems of nonlinear equations. To solve those systems we show that the use of inexact simplified Newton methods is efficient. Linear systems of the simplified Newton method are solved approximately with a preconditioned linear iterative method. Sufficient conditions ensuring local convergence of the inexact simplified Newton method for general nonlinear equations are given. The preconditioner that we use is based on the W-transformation of the RK coefficients and on the block-LU decomposition of the simplified Jacobian after W-transformation. A new code based on those techniques, SPARK3, is shown to be effective on two problems; the first one is a linear convection-diffusion problem and the second one a reaction-diffusion problem.

This paper introduces a general technique for the construction of multistep methods capable of integrating, without local truncation error, homogeneous linear ODEs with constant coefficients, including those, in particular, that result in oscillatory solutions. Moreover, these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies, even confluent ones that occur from nonhomogeneous terms in the differential equation. Our procedure allows the derivation of many of the existing codes with similar properties, as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency. The properties of the methods are studied within a general framework, and numerical examples are presented. These demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes, particularly when integrating the class of equations with perturbed oscillatory solutions. The methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known, as occurs in satellite orbit propagation. The underlying ideas have already been applied to the improvement of some Chebyshev methods that are not multistep.

A system of nonlinear partial differential equations is considered. The finite element scheme of Galerkin type is developed to get the semidiscrete approximations of the solutions. Stability results for the corresponding system of ordinary differential equations are shown. A sufficient criterion for the stability of the zero dynamics of an associated closed loop control system is verified. An application related to a spatially one-dimensional fixed-bed bioreactor is given.

The transfer of boundary conditions for ordinary differential equations developed by Abramov [Zh. Vychisl. Mat. Mat. Fiz., 1 (1961), pp. 542--545] is a stable method for representing the solution spaces of linear boundary value problems. Instead of boundary value problems, matrix-valued initial value problems are solved. When integrating these differential equations, the inner independence of the columns of the solution matrix and, hence, of the solutions of the resulting linear system of equations, remains valid. Balla and Marz have generalized Abramov's transfer for homogenized index-1 differential-algebraic equations [SIAM J. Numer. Anal., 33 (1996), pp. 2318--2332.] In this article, a direct version of the Abramov transfer for inhomogeneous linear $\mbox{index-1}$ differen-\linebreak[4]tial-algebraic equations is developed and the numerical stability of this method is proved.

We investigate the implementation of diagonally implicit multistage integration methods (DIMSIMs). The implementation issues addressed are the local error estimation, changing stepsize using the Nordsieck technique, and the construction of continuous interpolants. Numerical experiments with a method of order three indicate that the error estimates that have been constructed are very reliable in both a fixed and a variable stepsize environment.

We analyze in detail the growth with time (of the coefficients of the asymptotic expansion) of the error in the numerical integration with one-step methods of periodic solutions of systems of ordinary differential equations. Variable stepsizes are allowed. We successively consider "general," Hamiltonian, and reversible problems. For Hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated, numerical methods with relevant geometric properties (symplecticness, energy-conservation, reversibility) are proved to have better error growth than "general" methods.

In this paper we consider a mathematical model motivated by patterned growth of bacterial cells. The model is a system of differential equations that consists of two subsystems. One is a system of ordinary differential equations and the other is a reaction-diffusion system. An alternating-direction implicit (ADI) method is derived for numerically solving the system. The ADI method given here is different from the usual ADI schemes for parabolic equations due to the special treatment of nonlinear reaction terms in the system. Stability and convergence of the ADI method are proved. We apply these results to the numerical solution of a problem in microbiology.

In the numerical solution of initial value ordinary differential equations, to what extent does local error control confer global properties? This work concentrates on global steady states or fixed points. It is shown that, for systems of equations, spurious fixed points generally cease to exist when local error control is used. For scalar problems, on the other hand, locally adaptive algorithms generally avoid spurious fixed points by an indirect method---the stepsize selection process causes spurious fixed points to be unstable. However, problem classes exist where, for arbitrarily small tolerances, stable spurious fixed points persist with significant basins of attraction. A technique is derived for generating such examples.

This paper addresses the problems of detecting Hopf bifurcations in systems of ordinary differential equations and following curves of Hopf points in two-parameter families of vector fields. The established approach to this problem relies upon augmenting the equilibrium condition so that a Hopf bifurcation occurs at an isolated, regular point of the extended system. We propose two new methods of this type based on classical algebraic results regarding the roots of polynomial equations and properties of Kronecker products for matrices. In addition to their utility as augmented systems for use with standard Newton-type continuation methods, they are also particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension.

In this paper, the concept of Abramov's method for transferring boundary conditions posed for regular ordinary differential equations (ODEs) is applied to index-1 differential algebraic equations (DAEs). Having discussed the reduction of inhomogeneous problems to homogeneous ones and analyzed the underlying ideas of Abramov's method, we consider boundary value problems for index-1 linear DAEs both with constant and varying leading matrices. We describe the relations defining the subspaces of solutions satisfying the prescribed boundary conditions at one end of the interval. The index-1 DAEs which realize the transfer are given and their properties are studied. The results are reformulated for inhomogeneous index-1 DAEs as well.

Stochastic differential equations (SDEs) represent physical phenomena dominated by stochastic processes. As for deterministic ordinary differential equations (ODEs), various numerical schemes are proposed for SDEs. In this paper we study the stability of numerical schemes for scalar SDEs with respect to the mean-square norm, which we call $MS$-stability. We will show some figures of the $MS$-stability domain or regions for some numerical schemes and present numerical results which confirm it. This notion is an extension of absolute stability in numerical methods for ODEs.

In solving differential equations using a finite stepsize $h$, an error $Eh^{p+1}$ is generated at each step and propagates. This phenomenon is treated as a dynamical process, where $h(t)$ is controlled to optimize a properly defined performance index. Applying the variational principle, optimal stepsize is found to be proportional to $(E\psi)^{-\frac{1}{p+1}}$, where $E$ is the error generation coefficient and $\psi$ is the adjoint function of the error variable. This means that conventional adaptive control strategies depending on local information $(E)$ only are not optimal in general, except for two special cases. The theory is applied to three test problems for two figures of merit. The method is compared with several conventional strategies.

In this paper we consider diagonally split Runge–Kutta methods for the numerical solution of initial value problems for ordinary differential equations. This class of numerical methods was recently introduced by Bellen, Jackiewicz, and Zennaro [SIAM J. Numer. Anal., 31 (1994), pp. 499–523], and comprises the well-known class of Runge–Kutta methods. Their results strongly indicate that diagonally split Runge-Kutta methods break the order barrier $p \leq 1$ for unconditional contractivity in the maximum norm. In this paper we investigate the effect of the requirement of unconditional contractivity in the maximum norm on the accuracy of a diagonally split Runge–Kutta method. Besides the classical order p, we deal with an order of accuracy r which is relevant to the case where the method is applied to dissipative initial value problems that are arbitrarily stiff. We show that if a diagonally split Runge–Kutta method is unconditionally contractive in the maximum norm, then it has orders p, r which satisfy $p \...

Explicit Runge–Kutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations. So far, several Runge–Kutta pairs of order 6(5) have appeared in the literature. These pairs use 8 function evaluations per step and belong to certain families of solutions of a set of 54 nonlinear algebraic equations in 44 or 45 coefficients, depending on the use of the FSAL (first stage as last) device. These equations form a set of necessary and sufficient conditions that a 6(5) Runge–Kutta pair must satisfy. The solution of the latter is achieved by employing various types of simplifying assumptions. In this paper we make use of the fact that all these families of pairs satisfy a common set of simplifying assumptions. Using only these simplifying assumptions we define a new family of 6(5) Runge–Kutta pairs. Its main characteristic, which is also a property that no other known family shares, is that all of its nodes (except the last one, which eq...

We analytically show the exponential convergence of the sinc method applied to fourth-order ordinary differential equations (ODEs). Linear numerical examples are included to confirm our analytical results. The solution obtained with the sinc method is compared with the one obtained with a second- and a sixth-order finite difference scheme and with the ODE boundary-value solver COLNEW in the case of the linear equations. We then compute the steady-state solution of the one-dimensional Cahn–Hilliard equation to illustrate our results.

A general class of two-step Runge–Kutta methods that depend on stage values at two consecutive steps is studied. These methods are special cases of general linear methods introduced by Butcher and are quite efficient with respect to the number of function evaluations required for a given order. General order conditions are derived using the approach proposed recently by Albrecht, and examples of methods are given up to the order 5. These methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments.

In this paper we present a technique for the numerical approximation of a branch of invariant tori of finite-dimensional ordinary differential equations systems. Our approach is a discrete version of the graph transform technique used in analytical work by Fenichel [Indiana Univ. Math. J., 21 (1971), pp. 193–226]. In contrast to our previous work [L. Dieci, J. Lorenz, and R. D. Russell, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 607–647], the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori, but determines and updates such a coordinate system during a continuation process. We give general convergence results for the method and present its algorithmic description. We also show how the method performs on two physically important nonlinear problems, a system of two coupled oscillators and the forced van der Pol oscillator. In the latter case, we discuss some modifications needed to approximate an invariant curve for the Poincare map.

The author analyzes a finite element method for the integration of initial value problems in ordinary differential equations. General and contractive problems are treated, and quasi-optimal a priori and a posteriori error bounds obtained in each case. In particular, good results are obtained for a class of stiff dissipative problems. These results are used to construct a rigorous and robust theory of global error control. The author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error. In conclusion, the properties of the error control are exhibited in a series of numerical experiments.

The paper presents a new approach to the numerical solution of the Euler–Lagrange equations based upon the reduction of the problem to a second-order ordinary differential equation (ODE) on the constraint manifold. The algorithm guarantees that the constraints are automatically satisfied and requires a minimal number of evaluations of second-order derivative terms. In fact, second-order derivatives are involved only through the second fundamental tensor of the constraint manifold. This tensor may be computed either explicitly when second derivatives are available or via an approximation procedure with excellent accuracy. Examples are given along with comparisons with state-of-the-art software.

In this contribution, the author considers the one-block methods designed by Sommeijer, Couzy, and van der Houwen for the purpose of solving ordinary differential equations (ODEs) on a parallel computer. The author also derives a new set of order conditions, studies the stability, and exhibits a new class of parallel methods which contains L-stable schemes up to order eleven.

A posteriors error estimates for semi- and fully discrete finite element methods using a pth degree polynomial basis are considered for nonlinear parabolic equations. The error estimates are obtained by solving local parabolic or elliptic equations for corrections to the solution on each element using a $p + 1$st degree polynomial, which is zero at the nodes. Singly implicit Runge–Kutta (SIRK) and backward difference formula (BDF) methods are considered in the fully discrete case. Local errors are defined in an analogous manner to the respective methods for ordinary differential equations. The a posteriors error estimates are shown to converge to the local errors, even in case the local correction is missing from the nonlinear term. Computational examples compare and verify the theoretical results.

This paper deals with the relation between differential/algebraic equations (DAEs) and certain stiff ODEs and their respective discretizations by implicit Runge–Kutta methods. For that purpose for any DAE a singular perturbed ODE is constructed such that the DAE is its reduced problem and the solution of the ODE converges in some sense to that of the DAE. Thus the DAE can be interpreted as an infinitely stiff ODE. An analysis of the discretization error of this singular perturbed system gives insight into the relationship of order-reduction phenomena observed for stiff ODEs to that for DAEs. Analysis of a general class of singularly perturbed problems and their discretizations is not attempted; however, the technique of treating singularly perturbed problems and DAEs in a unified way is new and can possibly be applied to other systems and their discretizations as well. Since asymptotic expansions are not used, but an approach similar to the ones used in B-convergence theory is applied, one can derive erro...

This note gives a simple example of a superconvergence phenomenon which holds close to, but not quite up to, boundaries. It may thus serve to delineate what can be expected from any general theory of superconvergence.Consider a two-point boundary value problem for a second-order elliptic ordinary differential equation. Using the Galerkin method on a uniform mesh of size h with splines of odd polynomial degree and interelement continuity $\mathcal{C}^1 $ or higher, superconvergence to one extra order of the first derivative occurs at meshpoints and midpoints, provided we keep a distance $Ch\ln ({1 / h})$ away from the endpoints of the basic interval. An explicit example with Hermite cubics shows that this result is sharp. A numerical illustration is also given.

In this paper the problem of direct numerical integration of differential Riccati equations (DREs) and some related issues are considered. The DRE is an expression of a particular change of variables for a linear system of ordinary differential equations. The error that an approximate solution of the DRE induces on the original variables of the system is considered, and it is related to geometrical properties of the system itself. Sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system. Nonstiff and stiff DREs of unsymmetric and symmetric type are considered. A useful matrix interpretation is given for many integration schemes (such as the backward differentiation formulas, BDF), when applied to the DRE. This allows the matrix structure of the problem to be exploited. In particular, for stiff DREs, the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform...

Two numerical methods—one continuous and the other discrete—are proposed for solving inverse singular value problems. The first method consists of solving an ordinary differential equation obtained from an explicit calculation of the projected gradient of a certain objective function. The second method generalizes an iterative process proposed originally by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving inverse eigenvalue problems. With the geometry understood from the first method, it is shown that the second method (also, the method proposed by Friedland, Nocedal, and Overton for inverse eigenvalue problems) is a variation of the Newton method. While the continuous method is expected to converge globally at a slower rate (in finding a stationary point of the objective function), the discrete method is proved to converge locally at a quadratic rate (if there is a solution). Some numerical examples are presented.