Overview

Click this link to expand the page to include a web form.  Via web form you obtain the spatial discretisation of up to three coupled 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 slow mode coupled complex Ginzburg--Landau equation as an example (from Ispen and Sorensen 1999 ): $\frac{\partial u}{\partial t} =ru -(1+ia)|u|^2u +\frac{\partial^2u}{\partial x^2} +(1-ib)uw\,,$ $3\frac{\partial w}{\partial t} =-w -|u|^2 +2\frac{\partial^2w}{\partial x^2} \,.$ Sometimes referred to as a distributed slow-Hopf equation, this system is considered a normal form for oscillatory reaction-diffusion systems with a slow real mode w. Problem: the absolute function |u| is not analytic and cannot be manipulated simply in algebra. I remove this problem by the second of two alternatives: you could write two equations one for each of the real and imaginary parts of u; I prefer introducing the complex conjugate v = u*, then |u|2 = uv and the dynamical PDE for v is the complex conjugate of that for u: $\frac{\partial v}{\partial t} =rv -(1-ia)uv^2 +\frac{\partial^2v}{\partial x^2} +(1+ib)vw\,.$ These three coupled PDEs are given as an example.

The accompanying original web page empowers you to discretise a single PDE, such as Burgers' equation. Later versions will analyse coupled PDEs in two or more spatial dimensions, forced inhomogeneous PDEs and the discretisation near boundaries.

Read the overview of the theoretical support for our holistic discretisation.

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 coupled PDEs for analysis

Fill in the following fields for your coupled PDEs and its discretisation---assume the dependent fields u(x,t), v(x,t) and w(x,t) are a function of space x and time t. Use the syntax of Reduce for the algebraic expressions. Values of the fields for the slow mode coupled complex Ginzburg--Landau equation are listed in the third column as an example.

Description your coupled PDEs slow G-L
du/dt = RHS
Dominant dissipation factor on RHS df(u,x,2)
Coefficie nt of dominant dissipation factor on RHS (non-zero when u,v,w constant) 1
Dominant perturbing terms on RHS (linear or nonlinear, considered first order relative to u) r*u
Lesser perturbing terms on RHS (linear or nonlinear, considered second order relative to u) -(1+i*a)*u^2*v +(1-i*b)*u*w
dv/dt = RHS
Dominant dissipation factor on RHS df(v,x,2)
Coefficie nt of dominant dissipation factor on RHS (non-zero when u,v,w constant) 1
Dominant perturbing terms on RHS (linear or nonlinear, considered first order relative to v) r*v
Lesser perturbing terms on RHS (linear or nonlinear, considered second order relative to v) -(1-i*a)*u*v^2 +(1+i*b)*v*w
dw/dt = RHS
Dominant dissipation factor on RHS df(w,x,2)
Coefficie nt of dominant dissipation factor on RHS (non-zero when u,v,w constant) 2/3
Dominant perturbing terms on RHS (linear or nonlinear, considered first order relative to w) -w/3-u*v/3
Lesser perturbing terms on RHS (linear or nonlinear, considered second order relative to w) 0
expansion parameters
Width in space of the numerical stencil 5
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 2 for each order in the perturbation) decreasing
Print expressions with the following variables factored---this does not affect the analysis. h,r,a,b
Enter the magic word "w i n t e r" into then click

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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 with a large number of derivatives, even if each derivative is relatively low order.

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