Numerical stability of methods of lines for partial differential equations

The method of lines for the numerical treatment of partial differential equations is the technique, which consists in using finite differences for the approximation of derivatives with respect to all the independent variables except one, thus obtaining a set or a system of ordinary differential equations. These are then integrated on a digital computer by the use of the well-known numerical methods for such problems. For initial value type partial differential equations, the names of continuous space and continuous time are used to distinguish methods of lines where the remaining continuous independent variable is either a spatial coordinate x or the time t, respectively. Our interest in this paper will be restricted to the continuous time methods of lines for initial value problems. In these, one obtains a system of coupled ordinary differential equations, which is also of the initial value type. In addition to the classical implementation of numerical algorithms for the integration of such problems, the recent development of continuous simulation languages, such as CSMP [1] and CSSL [2], provides a means whereby certain of these algorithms are conveniently available as pre-programmed subroutines. This, plus the fact that considerable knowledge about the properties of the numerical integration of initial value problems in ordinary differential equations has been developed in recent years, has prompted a renewed interest in this area. The first description of methods of lines for the computer solution of partial differential equations seems to be that of Hartree [13], who was interested in the use of mechanical differential analyzers. More recently, Hicks and Wei [14] gave a concise description of the numerical implementation of continuous-time methods of lines for parabolic equations among others. Methods of lines have been one of the main tools in the application of electronic differential analyzers (analog and hybrid computers) to the solution of partial differential equations and a considerable literature has been devoted to this subject (see e.g., the references cited in [20]). It should also be mentioned that several authors have devoted their attention to methods of lines taken in a somewhat different context than that adopted here. in an often referred to paper, Rothe [18] used continuous-space methods of lines strictly as a concept to prove the existence of the solution of certain parabolic equations. The Russians are credited with the coinage of the name of "methods of lines". The reader is referred to the publications of Faddeyeva [7] and Budak [4] in this respect. A question, which has not been well analyzed to date, is that of the numerical stability of methods of lines. The prediction of numerical stability is, of course, of great importance in practice, and it is to that specific question that we address ourselves in this paper.Technical report DCS-TR-1