The composite Euler method for stiff stochastic differential equations

In this paper we present the composite Euler method for the strong solution of stochastic differential equations driven by d-dimensional Wiener processes. This method is a combination of the semi-implicit Euler method and the implicit Euler method. At each step either the semi-implicit Euler method or the implicit Euler method is used in order to obtain better stability properties. We give criteria for selecting the semi-implicit Euler method or the implicit Euler method. For the linear test equation, the convergence properties of the composite Euler method depend on the criteria for selecting the methods. Numerical results suggest that the convergence properties of the composite Euler method applied to nonlinear SDEs is the same as those applied to linear equations. The stability properties of the composite Euler method are shown to be far superior to those of the Euler methods, and numerical results show that the composite Euler method is a very promising method. (C) 2001 Elsevier Science B.V. All rights reserved.

Acute otitis media

A 3-year-old girl is brought to your office by her mother because she has a fever and complains that her ear hurts. She has no significant medical history. The child is not pleased to be in the physician's office and has been crying. Her mother explains that she developed a cold about 3 days ago with sniffles. Her temperature is 37.8 degreesC (100 degreesF), and the rest of the physical examination is completed with some difficulty. The only abnormalities are slight redness of the throat. a nose full of thick green mucus, and injected tympanic membranes. You wonder what findings other than red tympanic membranes should lead you to diagnose otitis media and also consider the recent controversy about whether to treat acute otitis media (AOM) with antibiotics.

Should watchful waiting be used more often for acute otitis media?

RECENT ANXIETY about the treatment of acute otitis media has been precipitated by a resistance to antibiotics by the common pathogens that can cause this infection.1, 2 The medical profession is facing an increasingly impotent option in the form of antibiotics, prompting physicians around the world to consider alternatives. In this issue of the ARCHIVES, Pichichero and Poole3 have undertaken a comprehensive study involving pediatricians and otorhinologists. The objectives were to assess their recognition of the physical findings of acute otitis media and their ability to perform myringotomy. The principal issue is the safety of performing myringotomy in children with acute otitis media. Because this is an office procedure in which a general anesthetic is not administered, the child is strapped to a papoose board and held down. Myringotomy is not without potential serious complications. The superior part of the middle ear cavity contains the ossicles and the chorda tympani branch . . . [Full Text of this Article]