The following does not work---it is under repair
Via this page you obtain the spatial discretisation of
a dynamical partial differential equation
) using dynamical systems theory.
The technique not only ensures consistency of the
discretisation, but remarkably theory ensures the
exponentially quick relevance of the discretisation at
finite grid spacing h. Theory also suggests the
numerical disretisation should have good stability
properties on a coarse spatial grid. I use a
generalised Burgers' equation as an example:
| || du |
| +u ||
|| d2u |
| +ru3 .
A companion page
s in one space dimension,
such as the complex Ginzburg--Landau equation. Later,
I may develop web pages interfacing tools for
discretisations in two or more spatial dimensions,
forced inhomogeneous PDE
s and the
discretisation near boundaries.
Submit your PDE for analysis
Fill in the following fields for your
and its discretisation---assume the
dependent field u(x,t) is a function of space x and
Use the syntax of Reduce for the algebraic expressions.
Values of the fields for the generalised
Burgers' equation are listed in the third column as an
Wait a minute or two
The analysis may take minutes after submitting. Be
patient. Perhaps make a cup of coffee while you wait.
I also generate a Matlab function implementing the
discretisation for integration by ODE45
or equivalent. I give no guarantee of the
performance of the discretisation, only that I have
endeavoured to analyse your PDE
according to the principles summarised in the supporting
Note: in this approach one must use a wide
enough stencil in order to represent nonlinear terms
with a large number of derivatives, even if each
derivative is relatively low order.
If you like this web page, please link to it
so others can find it more easily.