It has been the standard teaching of today that backward stability analysis is taught as absolute, just as in Newtonian physics time is taught as absolute time. We will prove that it is not true in general. It depends on algorithms. We will prove that forward and mixed stability analysis are absolutely invalid stability analysis in the sense that they have absolutely wrong reference points for ...

Inspired by the theory of modified equations (backward error analysis), a new approach to high-order, structure-preserving numerical integrators for ordinary differential equations is developed. This approach is illustrated with the implicit midpoint rule applied to the full dynamics of the free rigid body. Special attention is paid to methods represented as B-series, for which explicit formula...

In this paper we present error estimates for the ®nite element approximation of Poisson and modi®ed Helmholtz equations outside an obstacle or in a semi-in®nite strip in the plane. The ®nite element approximation is formulated on a bounded domain using a local approximate arti®cial boundary condition. In fact there is a sequence of local approximate boundary conditions for a given arti®cial bou...

We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow’s numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes, that are currently used, ...

We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow’s numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes, that are currently used, ...

The purpose of these notes is to give some examples illustrating how naive numerical approximations to PDE's may not work at all as expected. In addition, the following two important notions are introduced: (I) von Neumann stability analysis | helps identify when (and if) numerical schemes behave properly. (II) Arti cial viscosity | a tool in stabilizing numerical schemes. These notes should be...

In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: utt +uxxxx +u− |u|p−1u = 0 when p = 3 and p = 5. First we show the existence and the orbital stability of the standing wave solutions: u(x, t) = eφω(x). Next, we study the existence and linear stability of the traveling wave solutions: u(x, t) = φ(x + ct). For both types of ...

We construct curve counting invariants for a Calabi-Yau threefold Y equipped with a dominant birational morphism π : Y → X. Our invariants generalize the stable pair invariants of Pandharipande and Thomas which occur for the case when π : Y → Y is the identity. Our main result is a PT/DT-type formula relating the partition function of our invariants to the Donaldson-Thomas partition function in...

In this paper, we propose a new exact closed form mutual inductance equation for on-chip interconnects. We express the mutual inductance between two parallel rectangular conductors as a weighted sum of self-inductances. We do not place any restrictions on the alignment of the two parallel rectangular conductors. Moreover, they could be co-planar or reside on different layers. Most important, ex...