## TecML PDE

- V1.0
- overview

Different from the ODE TecML, PDE TecML can only describe single step discretization schemes. Consider you have a set of partial differential equation as follows.

dv/dt = a d^2v/dx^2;

In the FTCS (forward in time center in space) scheme of one dimensional case, we discretize above equation as follows.

dv/dt -> (v(n+1, x) – v(n, x)) / dt

d^2v/dx^2 -> (v(n, x+1) + v(n, x-1) – 2 v(n, x))/(dx)^2

We also have to add indices to each variable as follows.

v -> v(n, x)

Another important information to discretize PDE’s is boundary condition. - format

A TecML PDE file has three components: 1) variable definition part, 2) boundary condition and 3) differential term discretization equations.<tecml> <!-- variable definition --> <variable name="x" type="recurvar"/> ... <!-- equations --> <math> <!-- boundary condition equations --> <apply type = "boundary-cond" boundary-id = "1" location = "left"> ... </apply> <!-- differential term equations --> <apply> <eq/> <apply> <partialdiff/> <bvar> <ci>x</ci> </bvar> <ci> v </ci> </apply> ... </apply> ... </math> </tecml>

- description of each part
- variable definition part

One TecML variable correspond to one or more variables of a model file. Thus a TecML variable can be considered as a vector variable, however, in a TecML file, variables are denoted as scalar variables. In the variable definition part, every variable appears in the equation part are defined by “**variable**” tag with their “**name**” and “**type**” attributes. “**name**” attribute defines the variable name which must be unique in the file. “**type**” attribute defines the variable type where one of the following five types is specified for each variable.**type**meanings **recurvar**variables whose values are calculated by recurrence relation equations **arithvar**temporal variables (which can be mathematically eliminated) **constvar**constants **stepvar**variables for time step - equation part
- First, for all the terms which represent temporal or spatial differentials, original continuous and the corresponding discretized equations are defined.
- Next, the variables are described with corresponding discretized form. Usually, this part describes how the indices are supplied to each variable..
- All the equations are provided by MathML format.

- variable definition part
- XMLschema
- examples

Many PDE calculation schemes are provided as TecML PDE files.- FTCS forward time center space
- BTCS
- Crank-Nicolson

- Notes

- overview