mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a bi-directional delay line is a

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

technique used in

computer simulation Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...

for solving

ordinary differential equation 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 ...

s by converting them to

hyperbolic equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...

s. In this way an explicit solution scheme is obtained with highly robust numerical properties. It was introduced by Auslander in 1968. It originates from simulation of

hydraulic pipeline Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid coun ...

s where

wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative to ...

was studied. It was then found that it could be used as an efficient numerical technique for numerically insulating different parts of a simulation model in each times step. It is used in the

HOPSAN Hopsan is a free simulation environment for fluid and mechatronic systems, developed at Linköping University. Although originally developed for simulation of fluid power systems, it has also been adopted for other domains such as electric powe ...

simulation package (Krus ''et al.'' 1990). It is also known as the '' Transmission Line Modelling'' (TLM) from an independent development by Johns and O'Brian 1980. This is also extended to partial differential equations.

References

*D.M. Auslander, "Distributed System Simulation with Bilateral Delay Line Models", ''Journal of Basic Engineering, Trans. ASME p195-p200. June 1968.'' *P. B. Johns and M.O'Brien. "Use of the transmission line modelling (t.l.m) method to solve nonlinear lumped networks", ''The Radio Electron and Engineer.'' 1980. *P Krus, A Jansson, J-O Palmberg, K Weddfeldt.
Distributed Simulation of Hydromechanical Systems
. Presented at ''Third Bath International Fluid Power Workshop'', Bath, UK 1990. Numerical differential equations Numerical analysis {{mathanalysis-stub