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.

Numerical partial differential equations

Finite difference methods	Heat Equation and related: FTCS scheme · Crank–Nicolson method Hyperbolic: Lax–Friedrichs method · Lax–Wendroff method · MacCormack method · Upwind scheme · Other: Alternating direction implicit method · Finite-difference time-domain method

Finite volume methods	Godunov's scheme · High-resolution scheme · MUSCL scheme · AUSM · Riemann solver

Finite element methods	hp-FEM · Extended finite element method · Discontinuous Galerkin method · Spectral element method · Meshfree methods · Mortar methods

Other methods	Spectral method · Pseudospectral method · Method of lines · Multigrid methods · Collocation method · Level set method · Boundary element method · Immersed boundary method · Analytic element method · Particle-in-cell · Isogeometric analysis

Domain decomposition methods	Schur complement method · Fictitious domain method · Schwarz alternating method · Additive Schwarz method · Abstract additive Schwarz method · Neumann–Dirichlet method · Neumann–Neumann methods · Poincaré–Steklov operator · Balancing domain decomposition · BDDC · FETI · FETI-DP

References

E. N. Sarmin, L. A. Chudov (1963), On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method, USSR Computational Mathematics and Mathematical Physics, 3(6), (1537–1543).

A. Zafarullah (1970), Application of the Method of Lines to Parabolic Partial Differential Equations With Error Estimates, Journal of the Association for Computing Machinery, 17 (2), 294-302.

J. G. Verwer, J. M. Sanz-Serna (1984), Convergence of method of lines approximations to partial differential equations , Computing, 33(3-4), 297-313.

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. ISBN 1584884231.

Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press. ISBN 0126241309.

Schiesser, W. E. (1994). Computational mathematics in Engineering and Applied Science: ODEs, DAEs and PDEs. CRC Press. ISBN 0849373735.
Schiesser, W. E. and G. W. Griffiths (2009). A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press. ISBN 9780521519861.

Subramanian, V.R. and R.E. White (2004). Semianalytical method of lines for solving elliptic partial differential equations, Chemical Engineering Science, 59, 781-788.

External links

The Method Of Lines (MOL)

Vande Wouwer, Alain (2001). Adaptive Method of Lines. Chapman & Hall/CRC. ISBN 158488231X.