Difference schemes of optimum degree of implicitness for a family of simple ODEs with blow-up solutions

✍ Scribed by Rita Meyer-Spasche

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 748 KB
• Volume: 97
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(98)00107-1

✦ Synopsis

It has been speculated that the linearly implicit linearized trapezoidal rule should be especially useful for computing blow-up solutions since it features discrete blow-up. We study the question of how implicit a scheme should be from several different points of view: we discuss further the properties of the linearized trapezoidal and of its discrete blowup; we give exact schemes for a family of ODEs with polynomial nonlinearity, and that shows 'the optimum degree of implicitness' they need; we show that other standard schemes (the trapezoidal rule itself and the implicit midpoint rule) are exact on certain differential equations; we compare for several schemes and different nonlinearities the size of the leading error terms; and we briefly discuss two ways of adapting the degree of implicitness of a scheme to a given differential equation.