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# Method of lines

The method of lines (MOL, NMOL, NUMOL) (Schiesser, 1991; Hamdi, et al., 2007; Schiesser, 2009 ) is a technique for solving partial differential equations (PDEs) in which all but one dimension is discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of ODEs and DAEs, to be used. A large number of integration routines have been developed over the years in many different programming languages, and some have been published as open source resources; see for example Lee and Schiesser (2004).

The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s Sarmin and Chudov. Many papers discussing the accuracy and stability of the method of lines for various types of partial differential equations have appeared since (for example Zafarullah or Verwer and Sanz-Serna). W. E. Schiesser of Lehigh University is one of the major proponents of the method of lines, having published widely in this field.

## Application to elliptical equations

MOL requires that the PDE problem is well-posed as an initial value (Cauchy) problem in at least one dimension, because ODE and DAE integrators are initial value problem (IVP) solvers.

Thus it cannot be used directly on purely elliptic equations, such as Laplace's equation. However, MOL has been used to solve Laplace's equation by using the method of false transients (Schiesser, 1991; Schiesser, 1994). In this method, a time derivative of the dependent variable is added to Laplace’s equation. Finite differences are then used to approximate the spatial derivatives, and the resulting system of equations is solved by MOL. It is also possible to solve elliptical problems by a semi-analytical method of lines (Subramanian, 2004). In this method the discretization process results in a set of ODE's that are solved by exploiting properties of the associated exponential matrix. For a sample code, visit http://www.maple.eece.wustl.edu.

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