Overview
Click this link to expand the page to include a web form.
Via the web form you obtain the spatial discretisation of a
dynamical partial differential equation (PDE)
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
indicates the numerical discretisation should have good
stability properties on a coarse spatial grid. I use a
generalised Burgers' equation as an example:
\[\frac{\partial u}{\partial t} +u\frac{\partial u}{\partial x}
=\frac{\partial^2u}{\partial x^2} +ru^3.\]
A companion web page discretises
several coupled PDEs 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
PDEs and the discretisation near boundaries.
Since about 2016 there is a huge research endeavour to use
machine learning and artificial intelligence to achieve the
same results that this web page does for you algebraically,
and has been doing for nearly twenty years, and has the
assurance of well developed systematic mathematical theory.
Submit your PDE for analysis
Fill in the following fields for your PDE and its discretisation---assume the dependent field u(x,t) is a function of space x and time t. 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 example.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 theory. In this approach one must use a wide enough stencil in order to represent nonlinear terms that have 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.