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Stable and accurate interface conditions based on the SAT penalty method are derived for the linear advection-diffusion equation. The conditions are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator. We focus on high-order finite-difference operators that satisfy the summation-by-parts (SBP) property.We prove that stability is a natural consequence of the SBP operators used in conjunction with the new, penalty type, boundary conditions. In addition, we show that the interface treatments are conservative. The issue of the order of accuracy of the interface boundary conditions is clarified. New finite-difference operators of spatial accuracy up to sixth order are constructed which satisfy the SBP property. These finite-difference operators are shown to admit design accuracy ( pth-order global accuracy) when ( p − 1)th-order stencil closures are used near the boundaries, if the physical boundary conditions and interface conditions are implemented to at least pth-order accuracy. Stability and accuracy are demonstrated on the nonlinear Burgers' equation for a 12-subdomain problem with randomly distributed interfaces.

The small guanine nucleotide (GTP)-binding protein Rac regulates mitogen-induced cytoskeletal changes and c-Jun amino-terminal kinase (JNK), and its activity is required for Ras-mediated cell transformation. Epistatic analysis placed Rac as a key downstream target in Ras signalling; however, the biochemical mechanism regulating the cross-talk among these small GTP-binding proteins remains to be elucidated. Eps8 (relative molecular mass 97,000) is a substrate of receptors with tyrosine kinase activity which binds, through its SH3 domain, to a protein designated E3b1/Abi-1. Here we show that Eps8 and E3b1/Abi-1 participate in the transduction of signals from Ras to Rac, by regulating Rac-specific guanine nucleotide exchange factor (GEF) activities. We also show that Eps8, E3b1 and Sos-1 form a tri-complex in vivo that exhibits Rac-specific GEF activity in vitro. We propose a model in which Eps8 mediates the transfer of signals between Ras and Rac, by forming a complex with E3b1 and Sos-1.

Abstract. High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-byparts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area.

Finite difference approximations of the second derivative in space appearing in, parabolic, incompletely parabolic systems of, and second order hyperbolic, partial differential equations are considered. If the solution is pointwise bounded, we prove that finite difference approximations of those classes of equations can be closed with two orders less accuracy at the boundary without reducing the global order of accuracy.This result is generalised to initial-boundary value problems with an mth order principal part. Then, the boundary accuracy can be lowered m orders.Further, it is shown that summation-by-parts operators with approximating second derivatives are pointwise bounded. Linear and nonlinear computations corroborates the theoretical results.

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Stability for nonlinear convection problems using centered difference schemes require the addition of artificial dissipation. In this paper we present dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.