On the numerical solution of elliptic partial differential equations by the method of lines

The method of lines for Poisson's equation with nonlinear or free boundary conditions

“…In Section 3 two numerical examples are presented; the first one is a type of radiation problem on a fixed domain which illustrates the ease with which nonlinear boundary conditions and multiple solutions are treated; the second example is a time discretized free boundary problem arising in electrochemical machining [3], [10]. In the last section we provide a new analysis of the convergence of the method of lines for Poisson's equation on an irregular domain which extends the results of [5] and which supports the practical recommendations of [4] for the application of the method of lines to elliptic problems.…”

“…Indeed, much emphasis has been placed on using analytic solutions, which severely limits the applicability of the method (see [2] and [6]), and on avoiding the instability usually present when a straightforward shooting method is applied [4]. Yet stable efficient two point boundary value solvers now exist which can greatly extend the scope of problems to which the method of lines is applicable (see, e.g.…”

“…Because of advances in the solution of ordinary differential equations this method has long been popular in the Soviet Union (see the review paper [6] and the exposition in [2]). A more recent detailed discussion of the computational aspects of the method of lines for a variety of linear and nonlinear elliptic problems may be found in [4] while a theoretical analysis of the convergence of the by lines approximation for certain nonlinear model problems is given in [13]. Most discussions, e.g.…”

“…Most discussions, e.g. [4] and [13], assume that the problem is posed for a rectangular domain. Irregular domains which are not readily mapped onto rectangles (a situation commonly arising when the boundary is not known a priori) pose special problems because the method of lines approximation now represents a system of multipoint (possibly free) boundary problems.…”

Lumped-parameter representation of fluid-system transients

“…The MOL appears in text books on numerical solution of partial differential equations (Ames, 1977;Ortega and Poole, 1981;Rektorys, 1982;Holt, 1984) but is most usually applied to parabolic problems (Jones et al 1972;Anderson et al, 1984). However, it has also been applied to fluid transients, both for St Venant unsteady open-channel flow (Nougaro et al, 1967;Collins andFersht, 1968, Vichnevetsky, 1969;Oosterveld and Adamowski, 1976) and waterhammer (Galler and Westervelt, 1961;Wood, 1967;Thirriot et al, 1969;Maudsley, 1984) as well as for gas networks (Osiadacz, 1987), oil transport pipelines (Digernes, 1980), surge tank design (Anderson, 1983), internal combustion engine intake and exhaust systems (Lakshminarayanan et al, 1979), etc. The present objective is to examine the application of the MOL to hyperbolic fluid transient models, first to determine a fomulation that will give reliable results (in comparison with the MOC) and second to demonstrate its applicability to conventional analysis of stability (eg compressor surging).…”