Overview
I use a slow mode coupled complex Ginzburg--Landau equation as an example (from Ispen and Sorensen 1999 ): ∂u∂t=ru−(1+ia)|u|2u+∂2u∂x2+(1−ib)uw, 3∂w∂t=−w−|u|2+2∂2w∂x2. 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: ∂v∂t=rv−(1−ia)uv2+∂2v∂x2+(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.
If you like this web page, please link to it so others can find it more easily.