Overview
Via this page 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 GinzburgLandau equation as an example (from Ispen and Sorensen 1999):

Sometimes referred to as a distributed slowHopf equation, this system is considered a normal form for oscillatory reactiondiffusion 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:

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.
Submit your coupled PDEs for analysis
Fill in the following fields for your coupled PDEs and its discretisationassume 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 GinzburgLandau equation are listed in the third column as an example.
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.
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