Holistic discretisation of dynamical Partial Differential Equations

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Overview

Via this page 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 suggests the numerical disretisation should have good stability properties on a coarse spatial grid. I use a generalised Burgers' equation as an example:

du dt +u du dx = d2u dx2 +ru3 .

A companion page discretises 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.

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) df(u,x,4) df(u,x,6)	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 5 7 9	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)	2 3 4 decreasing	3
Print expressions with the following variables factored---this does not affect the analysis.		h,r

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.

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.