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.

Description your PDE is du/dt = RHS Burgers'
Dominant dissipation factor on RHS df(u,x,2)
Coefficient of dominant dissipation factor on RHS (non-zero when u=constant) 1
Dominant perturbing terms on RHS (linear or nonlinear, considered first order relative to u) -u*df(u,x)
Lesser perturbing terms on RHS (linear or nonlinear, considered second order relative to u) +r*u^3
Width in space of the numerical stencil 3
Order of error of the discretisation in the perturbations relative to u (the often useful decreasing option means for the perturbing terms the stencil width is reduced by two for each order in the perturbation) 3
Print expressions with the following variables factored---this does not affect the analysis. h,r
Enter the magic word "s p r i n g" into then click

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.

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