Method of Lines MOL and Differential Algebraic Equations DAE

Wu and White [577] have described a new method that is reminiscent of the earher work of Kimble and White [338] but makes use of the Hermitian method (that is, using derivatives) to achieve higher-order solutions for several concentration rows at a time. They also suggest, but do not demonstrate, the use of their new scheme as a possible start-up for BDF. The reader is referred to their paper for details. 

The Method of Lines or MOL is not so much a particular method as a way of approaching numerical solutions of pdes. It is described well by Hartree [295] as the replacement of the second-order (space) derivative by a finite difference that is, leaving the first (time) derivative as it is, thus forming from, say, the diffusion equation a set of ordinary differential equations, to be solved in an unspecified manner. Thus, a system such as (9.17) on page 151, can be written in the general vector-matrix form 

The method has a long history. The name MOL seems to have become estabhshed around 1960. Before this, various authors either used the word line [254] or expressions like on certain lines [330] or a description of the idea. In the book by Kantorovich and Krylov [330], there is a reference to a 1934 paper [329]. It is also cited by Liskovets [366] as a source paper, along with Rothe (1930) [475], who might be the first. Hartree and Womersley [296] use, in their summary, the words approximating by use of finite intervals in one variable, and integrating exactly in the other variable . The book by Schiesser [497] is the standard work now (he calls the method NUMOL, for numerical method of hnes). Electrochemical use of MOL has been sparse. Lemos and coworkers [357, 359, 360] have investigated the method, using 

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Method of Lines (MOL) and Differential Algebraic Equations (DAE) 165 Method of Wu and White... 

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