Numerical method of lines
Encyclopedia
The method of lines (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.
) 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.
Open source
The term open source describes practices in production and development that promote access to the end product's source materials. Some consider open source a philosophy, others consider it a pragmatic methodology...
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
Lehigh University
Lehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. It was established in 1865 by Asa Packer as a four-year technical school, but has grown to include studies in a wide variety of disciplines...
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 (CauchyCauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...
) problem in at least one dimension, because ODE and DAE integrators are initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
(IVP) solvers.
Thus it cannot be used directly on purely elliptic equations, such as Laplace's equation
Laplace's equation
In mathematics, 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:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
. 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.