On resolvent conditions and stability estimates

As many numerical processes for time discretization of evolution equations can be formulated as analytic mappings of the generator, they can be represented in terms of the resolvent. To obtain stability estimates for time discretizations, one therefore would like to carry known estimates on the resolvent back to the time domain. For different types of bounds of the resolvent of a linear operator, bounds for the norm of the powers of the operator and for their sum are given. Under similar bounds for the resolvent of the generator, some new stability bounds for one-step and multistep discretizations of evolution equations are then obtained.     

Remarks on Picard-Lindelöf iteration

The paper discusses Picard-Lindelöf iteration for systems of autonomous linear equations on finite intervals, as well as its numerical variants. Most of the discussion is under a model assumption which roughly says that the coupling terms are of moderate size compared with the slow time scales in the problem. It is shown that the speed of convergence is quite independent of the step sizes already for very large time steps. This makes it possible to design strategies in which the mesh gets gradually refined during the iteration in such a way that the iteration error stays essentially on the level of discretization error.This research was supported by the Academy of Finland and by the Institute for Mathematics and its Applications (IMA, Minneapolis with funds provided by NSF). Ulla Miekkala visited IMA with a grant from Tekniikan edisstämissäätiö and carried out the numerical tests.     

Graph properties for splitting with grounded Laplacian matrices

With any undirected, connected graphG containing no self-loops one can associate the Laplacian matrixL(G). It is also the (singular) admittance matrix of a resistive network with all conductances taken to be unity. While solving the linear system involved, one of the vertices is grounded, so the coefficient matrix is a principal submatrix ofL which we will call the grounded Laplacian matrixL ₁. In this paper we consider iterative solutions of such linear systems using certain regular splittings ofL ₁ and derive an upper bound for the spectral radius of the iteration matrix in terms of the properties of the graphG.This work was supported by the Academy of Finland     

A finite element — Capacitance method for elliptic problems on regions partitioned into subregions

Summary A method is given for the solution of linear equations arising in the finite element method applied to a general elliptic problem. This method reduces the original problem to several subproblems (of the same form) considered on subregions, and an auxiliary problem. Very efficient iterative methods with the preconditioning operator and using FFT are developed for the auxiliary problem.     

Numerical stability of descent methods for solving linear equations

Summary In this paper we perform a round-off error analysis of descent methods for solving a liner systemAx=b, whereA is supposed to be symmetric and positive definite. This leads to a general result on the attainable accuracy of the computed sequence {x _i } when the method is performed in floating point arithmetic. The general theory is applied to the Gauss-Southwell method and the gradient method. Both methods appear to be well-behaved which means that these methods compute an approximationx _i to the exact solutionA ^–1 b which is the exact solution of a slightly perturbed linear system, i.e. (A+A)x _i =b, A of order A, where is the relative machine precision and · denotes the spectral norm.     

Some vector sequence transformations with applications to systems of equations

First, recursive algorithms for implementing some vector sequence transformations are given. In a particular case, these transformations are generalizations of Shanks transformation and the G-transformation. When the sequence of vectors under transformation is generated by linear fixed point iterations, Lanczos' method and the CGS are recovered respectively. In the case of a sequence generated by nonlinear fixed point iterations, a quadratically convergent method based on the -algorithm is recovered and a nonlinear analog of the CGS method is obtained.     

Basis function selection and preconditioning high degree finite element and spectral methods

This work considers the effect of the choice of finite element basis on the conditioning and iterative solution of the algebraic systems obtained using high degree finite elements (p-methods). This issue is fundamental to the performance ofp-methods and to the increased use of iterative solution techniques both for standard finite element analysis and also in adaptivep solutions. The central ideas also apply to high degree spectral methods (global expansion methods).     

A necessary condition forB-convergence of Runge-Kutta methods

Algebraic stability is a well-known property necessary forB-convergence of a Runge-Kutta method, provided its nodesc _i are such thatc _i – c _j wheneveri j. In this paper a slightly weaker property is established which becomes algebraic stability as soon asc _i – c _j , wheneveri j is excluded.Using this condition it is shown that the Lobatto IIIA methods with more than two stages cannot beB-convergent.This paper was written while the author was visiting the Rijksuniversiteit Leiden in the Netherlands, supported by the Netherlands Organization for Scientific Research (N.W.O.) and by an Erwin Schrödinger scholarship from the Fonds zur Förderung der wissenschaftlichen Forschung.     

Stability of backward Euler multirate methods and convergence of waveform relaxation

For a large class of traditional backward Euler multirate methods we show that stability is preserved when the methods are applied to certain stable (but not necessarily monotonic) non-linear systems. Methods which utilize waveform relaxation sweeps are shown to be stable and converge for certain monotonic systems.     

On the convergence of the multigrid method for a hypersingular integral equation of the first kind

Summary We present a multigrid method to solve linear systems arising from Galerkin schemes for a hypersingular boundary integral equation governing three dimensional Neumann problems for the Laplacian. Our algorithm uses damped Jacobi iteration, Gauss-Seidel iteration or SOR as preand post-smoothers. If the integral equation holds on a closed, Lipschitz surface we prove convergence ofV- andW-cycles with any number of smoothing steps. If the integral equation holds on an open, Lipschitz surface (covering crack problems) we show convergence of theW-cycle. Numerical experiments are given which underline the theoretical results.     

API stepsize control for the numerical solution of ordinary differential equations

A control-theoretic approach is used to design a new automatic stepsize control algorithm for the numerical integration of ODE's. The new control algorithm is more robust at little extra expense. Its improved performance is particularly evident when the stepsize is limited by numerical stability. Comparative numerical tests are presented.     

Krylov sequences of maximal length and convergence of GMRES

In most practical cases, the convergence of the GMRES method applied to a linear algebraic systemAx=b is determined by the distribution of eigenvalues ofA. In theory, however, the information about the eigenvalues alone is not sufficient for determining the convergence. In this paper the previous work of Greenbaum et al. is extended in the following direction. It is given a complete parametrization of the set of all pairs {A, b} for which GMRES(A, b) generates the prescribed convergence curve while the matrixA has the prescribed eigenvalues. Moreover, a characterization of the right hand sidesb for which the GMRES(A, b) converges exactly inm steps, wherem is the degree of the minimal polynomial ofA, is given. This work was supported by AS CR Grant A2030706. Part of the work was performed while the third author visited Instituto di Analisi Numerica (IAN CNR).     

Error growth analysis via stability regions for discretizations of initial value problems

This paper deals with numerical methods for the solution of linear initial value problems. Two main theorems are presented on the stability of these methods. Both theorems give conditions guaranteeing a mild error growth, for one-step methods characterized by a rational function ϕ(z). The conditions are related to the stability regionS={z:z∈ℂ with |ϕ(z)|≤1}, and can be viewed as variants to the resolvent condition occurring in the reputed Kreiss matrix theorem. Stability estimates are presented in terms of the number of time stepsn and the dimensions of the space. The first theorem gives a stability estimate which implies that errors in the numerical process cannot grow faster than linearly withs orn. It improves previous results in the literature where various restrictions were imposed onS and ϕ(z), including ϕ′(z)≠0 forz∈σS andS be bounded. The new theorem is not subject to any of these restrictions. The second theorem gives a sharper stability result under additional assumptions regarding the differential equation. This result implies that errors cannot grow faster thann ^β, with fixed β<1. The theory is illustrated in the numerical solution of an initial-boundary value problem for a partial differential equation, where the error growth is measured in the maximum norm.     

Efficient sixth order methods for nonlinear oscillation problems

A class of two-step (hybrid) methods is considered for solving pure oscillation second order initial value problems. The nonlinear system, which results on applying methods of this type to a nonlinear differential system, may be solved using a modified Newton iteration scheme. From this class the author has derived methods which are fourth order accurate,P-stable, require only two (new) function evaluations per iteration and have a true real perfect square iteration matrix. Now, we propose an extension to sixth order,P-stable methods which require only three (new) function evaluations per iteration and for which the iteration matrix is a true realperfect cube. This implies that at most one real matrix must be factorised at each step. These methods have been implemented in a new variable step, local error controlling code.     

Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods

In this paper it is shown that the local discretization error ofs-stage singly-implicit methods of orderp can be estimated by embedding these methods intos-stage two-step Runge-Kutta methods of orderp+1, wherep=s orp=s+1. These error estimates do not require any extra evaluations of the right hand side of the differential equations. This is in contrast with the error estimation schemes based on embedded pairs of two singly-implicit methods proposed by Burrage.The work of A. Bellen and M. Zennaro was supported by the CNR and MPI. The work of Z. Jackiewicz was supported by the CNR and by the NSF under grant DMS-8520900.     

Accuracy and stability of multistage multistep formulas

Summary We continue our research concerning relations between accuracy and stability of time discretizations. Here we concentrate, rather than on the size and shape of the stability region, on the obtainable accuracy when the number of parameters in the formulas is large.     

The rate of convergence of an approximate matrix factorization semi-direct method

Summary The definition of acceleration parameters for the convergence of a sparseLU factorization semi-direct method is shown to be based on lower and upper bounds of the extreme eigevalues of the iteration matrix. Optimum values of these parameters are established when the eigenvalues of the iteration matrix are either real or complex. Estimates for the computational work required to reduce theL ₂ norm of the error by a specified factor are also given.     

Contractivity of Runge-Kutta methods

In this paper we present necessary and sufficient conditions for Runge-Kutta methods to be contractive. We consider not only unconditional contractivity for arbitrary dissipative initial value problems, but also conditional contractivity for initial value problems where the right hand side function satisfies a circle condition. Our results are relevant for arbitrary norms, in particular for the maximum norm.For contractive methods, we also focus on the question whether there exists a unique solution to the algebraic equations in each step. Further we show that contractive methods have a limited order of accuracy. Various optimal methods are presented, mainly of explicit type. We provide a numerical illustration to our theoretical results by applying the method of lines to a parabolic and a hyperbolic partial differential equation.Research supported by the Netherlands Organization for Scientific Research (N.W.O.) and the Royal Netherlands Academy of Arts and Sciences (K.N.A.W.)     

Discrete mechanics and its application to the solution of the n-body problem

A theory of discrete mechanics is developed based on the results of D. Greenspan. Discrete dynamical equations in an inertial frame, in a coordinate system related to some material point, and in a rotating frame are given and the consistency, stability, and convergence of the methods are studied and some numerical examples presented.     

A note on conjugate gradient convergence

Summary. The one-dimensional discrete Poisson equation on a uniform grid with points produces a linear system of equations with a symmetric, positive-definite coefficient matrix. Hence, the conjugate gradient method can be used, and standard analysis gives an upper bound of ) on the number of iterations required for convergence. This paper introduces a systematically defined set of solutions dependent on a parameter , and for several values of , presents exact analytic expressions for the number of steps ) needed to achieve accuracy . The asymptotic behavior of these expressions has the form )} as and )} as . In particular, two choices of corresponding to nonsmooth solutions give , i.e., iteration counts independent of ; this is in contrast to the standard bounds. The standard asymptotic convergence behavior, , is seen for a relatively smooth solution. Numerical examples illustrate and supplement the analysis. Received August 30, 1995 / Revised version received January 23, 1996