Higher order fully discrete scheme combined withH 1-Galerkin mixed finite element method for semilinear reaction-diffusion equations

We first apply a first order splitting to a semilinear reaction-diffusion equation and then discretize the resulting system by anH ¹-Galerkin mixed finite element method in space. This semidiscrete method yields a system of differential algebraic equations (DAEs) ofindex one. Apriori error estimates for semidiscrete scheme are derived for both differential as well as algebraic components. For fully discretization, an implicit Runge-Kutta (IRK) methods is applied to the temporal direction and the error estimates are discussed for both components. Finally, we conclude the paper with a numerical example.     

Solution of Index 2 Implicit Differential-Algebraic Equations by Lobatto Runge-Kutta Methods

We consider the numerical solution of systems of index 2 implicit differential-algebraic equations (DAEs) by a class of super partitioned additive Runge–Kutta (SPARK) methods. The families of Lobatto IIIA-B-C-C^*-D methods are included. We show super-convergence of optimal order 2s–2 for the s-stage Lobatto families provided the constraints are treated in a particular way which strongly relies on specific properties of the SPARK coefficients. Moreover, reversibility properties of the flow can still be preserved provided certain SPARK coefficients are symmetric.     

Decoupling of Abstract DAEs

We consider linear and time-invariant abstract differential-algebraic equations (DAEs) which are equations of the form E (t) = Ax (t) + f (t), x (0) = x₀. x (·) and f (·) are functions with values in Hilbert spaces X and Z and the operator E: X → Z is assumed to be bounded, whereas A is closed and defined on some dense subspace D (A). Motivated by the Kronecker normal form for the finite dimensional case, the decoupling of abstract DAEs is investigated. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear differential-algebraic equations with properly stated leading term: Regular points

We consider in this work linear, time-varying differential-algebraic equations (DAEs) of the form A(t)(D(t)x^′(t))+B(t)x(t)=q(t) through a projector approach. Our analysis applies in particular to linear DAEs in standard form E(t)x^′(t)+F(t)x(t)=q(t). Under mild smoothness assumptions, we introduce local regularity and index notions, showing that they hold uniformly in intervals and are independent of projectors. Several algebraic and geometric properties supporting these notions are addressed. This framework is aimed at supporting a complementary analysis of so-called critical points, where the assumptions for regularity fail. Our results are applied here to the analysis of a linear time-varying analogue of Chua's circuit with current-controlled resistors, displaying a rich variety of indices depending on the characteristics of resistive and reactive devices.     

Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of Hamiltonian ordinary differential equations by means of Newton-like iterations. We pay particular attention to time-symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For an s-stage IRK scheme used to integrate a \(\dim \)-dimensional system of ordinary differential equations, the application of simplified versions of Newton iterations requires solving at each step several linear systems (one per iteration) with the same \(s\dim \times s\dim \) real coefficient matrix. We propose a technique that takes advantage of the symplecticity of the IRK scheme to reduce the cost of methods based on diagonalization of the IRK coefficient matrix. This is achieved by rewriting one step of the method centered at the midpoint on the integration subinterval and observing that the resulting coefficient matrix becomes similar to a skew-symmetric matrix. In addition, we propose a C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the linear system and that takes special care in reducing the effect of round-off errors. We report some numerical experiments that demonstrate the reduced round-off error propagation of our implementation.     

Iterative Solution of SPARK Methods Applied to DAEs

In this article a broad class of systems of implicit differential–algebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge–Kutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the family of Lobatto IIIA-B-C-C^*-D coefficients, are crucial to deal properly with the presence of constraints and algebraic variables. A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. Inexact modified Newton iterations can be used to solve these systems. Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method. Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations. These transformations rely heavily on specific properties of the SPARK coefficients. A new truly parallelizable preconditioner is presented.     

Reversible methods of Runge-Kutta type for Index-2 DAEs

Summary. A new interpretation of Runge-Kutta methods for differential algebraic equations (DAEs) of index 2 is presented, where a step of the method is described in terms of a smooth map (smooth also with respect to the stepsize). This leads to a better understanding of the convergence behavior of Runge-Kutta methods that are not stiffly accurate. In particular, our new framework allows for the unified study of two order-improving techniques for symmetric Runge-Kutta methods (namely post-projection and symmetric projection) specially suited for solving reversible index-2 DAEs.Mathematics Subject Classification (1991): 65L05, 65L06     

Modified Lagrange–Galerkin methods of first and second order in time for convection–diffusion problems

We introduce modified Lagrange–Galerkin (MLG) methods of order one and two with respect to time to integrate convection–diffusion equations. As numerical tests show, the new methods are more efficient, but maintaining the same order of convergence, than the conventional Lagrange–Galerkin (LG) methods when they are used with either P ₁ or P ₂ finite elements. The error analysis reveals that: (1) when the problem is diffusion dominated the convergence of the modified LG methods is of the form O(h ^m+1 + h ² + Δt ^q ), q = 1 or 2 and m being the degree of the polynomials of the finite elements; (2) when the problem is convection dominated and the time step Δt is large enough the convergence is of the form O(\frach^m+1Dt+h²+Dt^q){O(\frac{h^{m+1}}{\Delta t}+h^{2}+\Delta t^{q})} ; (3) as in case (2) but with Δt small, then the order of convergence is now O(h ^m  + h ² + Δt ^q ); (4) when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter ν (x, t), so that when ν → 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem. Our error analysis shows that the conventional LG methods exhibit the same error behavior as the MLG methods but without the term h ². Numerical experiments support these theoretical results.     

On a criterion for the complete controllability of nonautonomous linear systems

We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ ⁿ , u ∈ U ⊆ ℝ ^m , with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria.     

Global analysis of continuous analogues of the Levenberg-Marquardt and Newton-Raphson methods for solving nonlinear equations

Summary Global analyses are given to continuous analogues of the Levenberg-Marquardt methoddx/dt=−(J ^t(x)J(x)+δI)⁻¹J^t(x)g(x), and the Newton-Raphson-Ben-Israel methoddx/dt=−J ⁺(x)g(x), for solving an over- and under-determined systemg(x)=0 of nonlinear equations. The characteristics of both methods are compared. Erros in some literature which dealt with related continous analogue methods are pointed out. The Institute of Statistical Mathematics     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

Robust Controllability of Linear Differential-Algebraic Equations with Unstructured Uncertainty

We consider the linear stationary systems of ordinary differential equations (ODEs) that are unsolvedwith respect to the derivative of the unknown vector-function and degenerate identically in the domain of definition. These systems are usually called differential-algebraic equations (DAEs). The measure of how a system of DAEs is unsolved with respect to the derivative is an integer which is called the index of the system of DAEs. The analysis is carried out under the assumption of existence of a structural form with separated differential and algebraic subsystems. We investigate the robust controllability of these systems (controllability in the conditions of uncertainty). The sufficient conditions for the robust complete and R-controllability of a system of DAEs with the indices 1 and 2 are obtained.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smooth Lyapunov Functions for Discontinuous Stable Systems

It has been proved that a differential system d x / d t = f(t, x) with a discontinuous right-hand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This paper focuses on the smoothness of such a Lyapunov function. An example of an (asymptotically) stable system for which there does not exist any (even weak) Lyapunov functions of class C ¹ is given. In the more general context of differential inclusions, the existence of a weak Lyapunov function of class C ¹ (or C ) is shown to be equivalent to the robust stability of some perturbed system obtained in introducing measurement error with respect to x and t. This condition is proved to be satisfied by most of the robustly stable systems encountered in the literature. Analogous results are given for the Lagrange stability. As an application to the study of the links between internal and external stability for control systems, an extension of a result by Bacciotti and Beccari is obtained by means of a smooth Lyapunov function associated with a robustly Lagrange stable system.     

Numerical stroboscopic averaging for ODEs and DAEs

The stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems dy/dt=f(y,t) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system dY/dt=F(Y) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step-sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LIPS algorithm of Kirchgraber and multirevolution methods. An error analysis is provided that indicates that the efficiency of all these techniques increases even further when combined with splitting integrators.     

Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

The asymptotic behaviour and stability properties are studied for a real two‐dimensional system x^′(t) = A(t)x (t) + B(t)x (θ (t)) + h (t, x (t), x (θ (t))), with a nonconstant delay t ‐ θ (t) ≥ 0. It is supposed that A,B and h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex‐valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov‐Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269 , No. 1, 278–300 (2002)] for two‐dimensional systems with a constant delay (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New large sets of t‐designs

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t‐designs. These are based on assembling systematically orbits of t‐homogeneous permutation groups in their induced actions on k‐subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m ? 2, 16·3^m ? 3) is obtained. This also provides the smallest known ν for a t‐(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001     

Upper bound on the coarsening rate for an epitaxial growth model

We study a specific example of energy‐driven coarsening in two space dimensions. The energy is ∫|??u|² + (1 ‐ | ?u|²)²; the evolution is the fourth‐order PDE representing steepest descent. This equation has been proposed as a model of epitaxial growth for systems with slope selection. Numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t^1/3, and the energy per unit area decays like t^‐1/3. We prove a weak, one‐sided version of the latter statement: The time‐averaged energy per unit area decays no faster than t^‐1/3. Our argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining (i) a dissipation relation, (ii) an isoperimetric inequality, and (iii) an ODE lemma. The interpolation inequality is new and rather subtle; our proof is by contradiction, relying on recent compactness results for the Aviles‐Giga energy. © 2003 Wiley Periodicals, Inc.     

Positive solution for asymptotically linear elliptic systems

In this paper, we show that semilinear elliptic systems of the form (1) possess at least one positive solution pair (u, v)∈H¹(?^N)×H¹(?^N), where λ and µ are nonnegative numbers, f(x, t) and g(x, t) are continuous functions on ?^N×? and asymptotically linear as t→+∞. Copyright © 2009 John Wiley & Sons, Ltd.     

Butcher's simplifying assumption for symplectic integrators

Implicit Runge-Kutta methods with vanishingM matrix are discussed for preserving the symplectic structure of Hamiltonian systems. The number of the order conditions independent of the number of stages can be reduced considerably for the symplectic IRK method through the analysis utilizing the rooted tree and the corresponding elementary differentials. Butcher's simplifying condition further reduces the number of independent order conditions.