The method of lines
(MOL, NMOL, NUMOL) ( Schiesser, 1991
; Hamdi, et al., 2007
; ) is a technique for solving partial differential equations
(PDEs) where all but one dimension is discretized. The resulting semi-discrete
problem is a set of ordinary differential equations
(ODEs) or differential algebraic equations
(DAEs) that is then integrated. A significant advantage of MOL is that it allows standard general purpose methods and software to be used that have been developed for the numerical integration of ODEs and DAEs. For the PDEs where it is suitable, MOL is an efficient solution method. 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)
It is necessary that the PDE problem be well-posed as an initial value (Cauchy) problem in at least one dimension. This is because ODE and DAE integrators are initial value problem (IVP) solvers. This generally rules out purely elliptic equations such as Laplace's equation, but leaves a large class of evolution equations that can be solved quite efficiently. 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 (Subramaniana, 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.
W. E. Schiesser of Lehigh University is one of the major proponents of the method of lines .
- Hamdi, S., W. E. Schiesser and G. W. Griffiths (2007), Method of lines, Scholarpedia, 2(7):2859.
- Lee, H. J. and W. E. Schiesser (2004). Ordinary and Partial Differential Equation Routines in C, C++, Fortran, Java, Maple and Matlab. CRC Press.
- Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press.
- Schiesser, W. E. (1994). Computational mathematics in Engineering and Applied Science: ODEs, DAEs and PDEs. CRC Press.
- Subramaniana, V.R. and R.E. White (2004). Semianalytical method of lines for solving elliptic partial differential equations, Chemical Engineering Science, 59, 781-788.