Special boundedness properties in numerical initial value problems

For Runge-Kutta methods, linear multistep methods and other classes of general linear methods much attention has been paid in the literature to important nonlinear stability properties known as total-variation-diminishing (TVD), strong stability preserving (SSP) and monotonicity. Stepsize conditions guaranteeing these properties were studied by Shu \& Osher (1988) and in numerous subsequent papers. Unfortunately, for many useful methods it has turned out that these properties do not hold. For this reason attention has been paid in the recent literature to the related and more general properties called total-variation-bounded (TVB) and boundedness. In the present paper we focus on stepsize conditions guaranteeing boundedness properties of a special type. These boundedness properties are optimal, and distinguish themselves also from earlier boundedness results by being relevant to sublinear functionals, discrete maximum principles and preservation of nonnegativity. Moreover, the corresponding stepsize conditions are more easily verified in practical situations than the conditions for general boundedness given thus far in the literature. The theoretical results are illustrated by application to the two-step Adams-Bashforth method and a class of two-stage multistep methods

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

In this paper nonlinear monotonicity and boundedness properties are analyzed for linear multistep methods. We focus on methods which satisfy a weaker boundedness condition than strict monotonicity for arbitrary starting values. In this way, many linear multistep methods of practical interest are included in the theory. Moreover, it will be shown that for such methods monotonicity can still be valid with suitable Runge-Kutta starting procedures. Restrictions on the stepsizes are derived that are not only sufficient but also necessary for these boundedness and monotonicity properties

Monotonicity conditions for multirate and partitioned explicit Runge-Kutta schemes

Multirate schemes for conservation laws or convection-dominated problems seem to come in two ¿avors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods. It will also be seen that the incompatibility of consistency and mass-conservation holds for ‘genuine’ multirate schemes, but not for general partitioned methods

An integral method for solving nonlinear eigenvalue problems

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least $k$ column vectors, where $k$ is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension $k$. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where $k$ is much smaller than the matrix dimension. We also give an extension of the method to the case where $k$ is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour