Line method

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The vertical line method (engl. Method of lines , MOL) is a method (to release parabolic ) partial differential equations , wherein all but one dimension (typically the time variable) discretized be. As a result of the discretization, instead of the original partial differential equation, a system of ordinary differential equations results, which can be treated with adequate means. The numerical version, also called “NMOL”, is of particular interest . In this case, the solution of the system of ordinary differential equations obtained through the discretization takes place, for example, through the use of one- step or multi-step methods , in particular Runge-Kutta methods . This fact already shows the limits of the possible uses of this method: In order to be able to use one- or multi-step methods, the problem that results after the discretization must represent a first-order initial value problem , which in turn means that the original problem is a first-order initial value problem in at least one variable have to be.

This method is opposed to the horizontal line method , which is better known under the name Rothe method (named after Erich Rothe ). The idea of ​​the Rothe method for parabolic initial value problems is to first perform a discretization with respect to time, in order to reformulate the problem directly to an initial value problem in the function space.

Vertical line method

The idea behind the (vertical) line method for parabolic initial boundary value problems is to first discretize a discretization in terms of the spatial variables and then to discretize the resulting problem in terms of time. In the case of a conformal approximation, let (see Sobolev spaces ), (see L p spaces ) and . The generalized problem of a parabolic differential equation now means: Find a with such that:

,

where is a constrained, V-elliptical bilinear shape on and .

If the spatial discretization is done with finite elements , then we get the discrete problem for (finite element function space ):

,

where an approximation of .

Now be a basis of and . Then the Galerkin equations for the discrete problem described above result:

,

with .

With this we get a differential equation of the form

,

where with , with and or and .

Horizontal line method (Rothe method)

We are going back from the generalized form

,

with and off. Then the time interval is divided into sub-intervals with the grid width. This time it is the cone function in time, that is, in the case of a temporal discretization with the grid points, applies

.

Then an approximation for is described by the Rothe function

.

Using the implicit Euler method , one now solves the location problem in every time step

,

whereby . The use of other integration methods is also possible; however, since the problems are mostly stiff, an implicit approach should be preferred.

literature

  • William E. Schiesser: The Numerical Method of Lines. Integration of partial differential equations. Academic Press, San Diego et al. 1991, ISBN 0-12-624130-9
  • William E. Schiesser: Computational mathematics in Engineering and Applied Science. ODEs, DAEs, and PDEs. CRC Press, Boca Raton FL et al. 1994, ISBN 0-8493-7373-5 .

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