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The method of lines (MOL, NMOL, NUMOL) is a technique for solving
False Transient Method of Lines - sample code
Numerical differential equations Partial differential equations {{applied-math-stub
The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
(PDEs) in which all but one dimension is discretized. By reducing a PDE to a single continuous dimension, the method of lines allows solutions to be computed via methods and software developed for the numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
of ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
(ODEs) and differential-algebraic systems of equations (DAEs). Many integration routines have been developed over the years in many different programming languages, and some have been published as open source
Open source is source code that is made freely available for possible modification and redistribution. Products include permission to use the source code, design documents, or content of the product. The open-source model is a decentralized sof ...
resources.
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. Many papers discussing the accuracy and stability of the method of lines for various types of partial differential equations have appeared since.
Application to elliptical equations
MOL requires that the PDE problem is well-posed as an initial value (Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
) problem in at least one dimension, because ODE and DAE integrators are initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or oth ...
(IVP) solvers. Thus it cannot be used directly on purely elliptic partial differential equations
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
where ...
, such as Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
. However, MOL has been used to solve Laplace's equation by using the ''method of false transients''. 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''. In this method, the discretization process results in a set of ODE's that are solved by exploiting properties of the associated exponential matrix.
Recently, to overcome the stability issues associated with the method of false transients, a perturbation approach was proposed which was found to be more robust than standard method of false transients for a wide range of elliptic PDEs.
References
External links
False Transient Method of Lines - sample code
Numerical differential equations Partial differential equations {{applied-math-stub