Holistic discretisation of a dynamical Partial Differential Equation

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.