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  • CR: 5.17
  • Springer  (19)
  • Oxford University Press
  • 1980-1984  (19)
  • 1983  (19)
Collection
Keywords
Publisher
  • Springer  (19)
  • Oxford University Press
Years
  • 1980-1984  (19)
Year
  • 1
    Electronic Resource
    Electronic Resource
    A semi-implicit mid-point rule for stiff systems of ordinary differential equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 373-398 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper introduces a new semi-implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations. The existence of a quadratic asymptotic expansion in terms of the stepsize is shown. Moreover, the new discretization is analyzed in the light of well-known stability models. The efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418331
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  • 2
    Electronic Resource
    Electronic Resource
    Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 345-371 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418330
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  • 3
    Electronic Resource
    Electronic Resource
    On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 1-13 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere $$\mathbb{C}$$ ). It is shown that the solutions of these difference equations are bounded forn≧0, uniformly for λh∈D.S is itself closed in $$\mathbb{C}$$ iff ∂S is free of cusps. The question is studed by means of contractivity analysis and a matrix theorem, derived from the matrix theorem of Kreiss.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400914
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  • 4
    Electronic Resource
    Electronic Resource
    Order and stepsize control in extrapolation methods (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 399-422 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper presents a new theory for joint order and stepsize control in extrapolation methods. This theory defines a locally optimal order that can be determined along any trajectory to be computed. In addition, Shannon's information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems. Extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non-stiff integration by the new device. Moreover, a significant increase in reliability, robustness, and portability of the extrapolation codes is achieved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418332
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  • 5
    Electronic Resource
    Electronic Resource
    Split linear multistep methods for the numerical integration of stiff differential systems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 299-310 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new approach to the problem of numerically integrating stiff differential systems is described. In this approach a linear multistep method (the basic method) is split into a kind of predictor-corrector scheme, where the predictor is also implicit. If this splitting is done in an appropriate manner, the modified method has considerably better stability properties than the basic method. As a result, splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties. In particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389575
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  • 6
    Electronic Resource
    Electronic Resource
    A boundary element method for a two-dimensional interface problem in electromagnetics (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 311-322 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 N 30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents Galerkin approximations for solutions of two dimensional interface problems by solving corresponding boundary integral equations. These are obtained by simple layer potential operators only. Due to the strong ellipticity of the integral equations the Galerkin procedure converges with optimal order. Smoothness of the given data implies high convergence rates for the layers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389576
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  • 7
    Electronic Resource
    Electronic Resource
    OnD-stability andB-stability (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 349-357 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper theD-stability properties of some classes of perturbed collocation methods are investigated. Among others we show the implicit Runge-Kutta methods described by Ehle-Chipman to beD-stable. We also give implicit Runge-Kutta methods that are bothD-stable andA-stable but notB-stable. Finally we indicate in what senseB-stability is stronger thanA-stability together withD-stability. The results of this paper are based on recent results of Nørsett and Wanner on perturbed collocation methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389579
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  • 8
    Electronic Resource
    Electronic Resource
    Error analysis for Laplace transform —Finite element solution of hyperbolic equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 55-62 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The finite element method with Laplace transform of time variable is proposed for the solution of hyperbolic equations. Error estimates in Hardy spaces of functions with values in Sobolev spaces are derived. Due to the isometric isomorphism of Hardy spaces with weighted Hilbert spaces these estimates are valid also for original formulations of hyperbolic equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396305
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  • 9
    Electronic Resource
    Electronic Resource
    Smoothing the extrapolated midpoint rule (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 165-175 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65LO5 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The extrapolated midpoint rule is a popular way to solve the initial value problem for a system of ordinary differential equations. As originally formulated by Gragg, the results are smoothed to remove the weak instability of the midpoint rule. It is shown that this smoothing is not necessary. A cheaper smoothing scheme is proposed. A way to exploit smoothing to increase the robustness of extrapolation codes is formulated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390211
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  • 10
    Electronic Resource
    Electronic Resource
    On calculating the eigenvalues of the finite Hill's differential equation (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 255-279 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L15 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Some methods for evaluating the characteristic exponents in connection with Newton's iteration are applied for solving the eigenvalue problem related to the finite Hill's differential equation or, in particular, Mathieu's equation. By using these methods a high accuracy is achieved, furthermore a complete error analysis, which yields rather realistic error bounds, is possible.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390215
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  • 11
    Electronic Resource
    Electronic Resource
    Monotone methods for periodic solutions of second order scalar functional differential equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 15-30 
    Details
    ISSN: 0945-3245
    Keywords: AMS: 65L10 ; 34C25 ; 34K10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An iterative method is presented which starting from a lower or from an upper periodic solution, provides a monotone sequence converging to a periodic solution of (1). With some restrictions on the growth off, the method extends to functional differential equations of type (1′). Two numerical examples with an “a posteriori” error analysis are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400915
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  • 12
    Electronic Resource
    Electronic Resource
    Approximation of pseudodifferential operators in absorbing boundary conditions for hyperbolic equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 51-64 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65N99 ; 35L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Engquist and Majda [3] proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. (In the case of the wave equation this boundary condition is valid at all frequencies.) Here, least-squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. It is shown that for the wave equation this approach leads to Kreiss well-posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of Taylor- and Padé-approximations [3, 4]. Numerical examples indicate that this method works even more effectively for hyperbolic systems. The least-squares approach may be used to generate the boundary conditions automatically.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400917
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  • 13
    Electronic Resource
    Electronic Resource
    On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 77-95 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N99 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Brakhage and Werner, Leis and Panich suggested to reduce the exterior Dirichlet boundary value problem for the Helmholtz equation to an integral equation of the second kind which is uniquely solvable for all frequencies by seeking the solution in the form of a combined double- and single-layer potential. We present an analysis of the appropriate choice of the parameter coupling the double- and single-layer potential in order to minimize the condition number of the integral operator.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400919
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  • 14
    Electronic Resource
    Electronic Resource
    On multi-grid methods for variational inequalities (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 65-76 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 N 20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400918
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  • 15
    Electronic Resource
    Electronic Resource
    Die maximale Konsistenzordnung von Differenzenapproximationen nichtnegativer Art (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 119-123 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Many difference methods for the numerical solution of elliptic boundary value problems lead to systems of linear equations whose matrices areM-matrices and which therefore have nonnegative inverses. In this paper it is shown, that these difference methods are at most consistent of second order.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400922
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  • 16
    Electronic Resource
    Electronic Resource
    On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 173-194 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395310
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  • 17
    Electronic Resource
    Electronic Resource
    Contractivity in the numerical solution of initial value problems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 271-290 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J10, 65L20, 65M10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|≦|U(0)| (t≧0). We call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations. A givenk-step method is said to be unconditionally contractive if for each stepsizeh〉0 the numerical process is contractive. In this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity. It turns out that unconditionally contractive methods are subject to an order barrierp≦1. Further the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behaviour of methods with an orderp〉1 as well. Most theoretical results in this paper are formulated for differential equations in arbitrary Banach spaces. Applications are given to numerical methods for solving ordinary as well as partial differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389573
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  • 18
    Electronic Resource
    Electronic Resource
    Stability of multistep-methods on variable grids (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 359-377 
    Details
    ISSN: 0945-3245
    Keywords: AMS 65L05 ; 65L07 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper is concerned with the stability of multistep methods for ordinary initial-value problems on grids with variable mesh-sizes. A necessary and sufficient condition for stability is given from which generalizations of recent results by Gear et al. and by Zlatev can be obtained as special cases. As an application the stability of the variable BDF-formulas is treated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389580
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  • 19
    Electronic Resource
    Electronic Resource
    Norm bounds for rational matrix functions (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 379-389 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L20 ; 15A60 ; 65M20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary If the field of values of a matrixA is contained in the left complex halfplaneH and a functionf mapsH into the unit disc then ∥f(A)∥2≦1 by a theorem of J.v. Neumann. We prove a theorem of this type, only the field of values ofA is used for functions which are absolutely bounded by one in only part ofH. An extension can be used to show norm-stability of single step methods for stiff differential equations. The results are applicable among others to several subdiagonal Padé approximations which are notA-stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389581
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