“Mediation-Only” Filings in the Delaware Court of Chancery: Can New Value Be Added by One of America’s Business Courts?

The following Essay by Vice Chancellor Leo Strine of the Delaware Court of Chancery advocates the enactment of legislation that authorizes the Court of Chancery to handle mediation-only cases. Such cases would be filed solely to invoke the aid of a Chancellor to mediate a business dispute between parties. By advocating this innovative dispute resolution option, the Essay embraces a new dimension of the American judicial role that allows American businesses to more efficiently solve complicated business controversies. The mediation-only device was conceived in 2001 by members of the Delaware judiciary, including Vice Chancellor Strine, in consultation with members of the Delaware Bar and the Administration of Delaware Governor Ruth Ann Minner. After this Essay was widely circulated to certain constituencies and presented at a symposium sponsored by the Duke Law Journal and the Institute for Law and Economic Policy (ILEP), legislation that contained the mediation-only device was drafted. In June 2003, with the full support of the Court of Chancery, Delaware Governor Minner secured passage of the legislation from Delaware\u27s General Assembly. The mediation-only device was enacted into law as 346 and 347 of Title 10 of the Delaware Code. To the Editors\u27 knowledge, this legislation is the first of its kind adopted in the United States

The Rate-Making Process in Property and Casualty Insurance—Goals, Technics, and Limits

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method, that we refer to as the multiple penalty technique, is similar to Davies relaxation and have similar areas of application. The method is proven, by energy methods, to be stable. We show how to apply this technique on the linearized Euler equations in two space dimensions, and that it reduces the errors in the computational domain

High-order finite difference approximations for hyperbolic problems : multiple penalties and non-reflecting boundary conditions

In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal

Spurious solutions for the advection-diffusion equation using wide stencils for approximating the second derivative.

A one dimensional steady-state advection-diffusion problem using summation-by-parts operators has been investigated. For approximating the second derivative, a wide stencil has been used, which has spurious, oscillating, modes for all mesh-sizes. We show that the size of the spurious modes are equal to the size of the truncation error for a stable approximation. The theoretical results are veried with numerical experiments

Long time error bounds for the wave equation on second order form

Temporal error bounds for the wave equation expressed on second order form is investigated. By using the energy method, we show that, with appropriate choices of boundary condition, the time and space derivative of the error is bounded even for long times. No long time bound on the actual error can be obtained, although numerical experiments indicate that such a bound exist

Constructing non-reflecting boundary conditions using summation-by-parts in time

In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments

A Stable and Accurate Davies-like Relaxation Procedure using Multiple Penalty Terms for Lateral Boundary Conditions

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method is similar to Davies relaxation technique used in the weather prediction community and have similar areas of application, but is also provably stable. In this paper, it is shown how this technique can be applied to the shallow water equations, and that it reduces the errors in the computational domain.Funding agencies: Swedish e-science Research Center (SeRC)</p

Constructing non-reflecting boundary conditions using summation-by-parts in time

In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments