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I'm just starting at scientific computing and I'm seeing that results can be generated merely by plotting the variables. Of course more variables lead to more axes/dimensions so e.g. a three variable system could be or would need to be modeled as a 3D plot.

What I was thinking though was e.g. a 2D case. Does visualization e.g. through some dummy variables (to add dimensions, i.e. to create a 3D visualization out of 2D variables) add anything to simple X-Y plots?

Or is visualization always linked to the dimensions?

Does visualization (e.g. an animation) add something more to the "generic plots"? Or is visualization often the same as the plot?

What about in the case of DEs, which may have multiple variables?

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Visualization, generally, is the science of translating data to insight via graphical representations. There are as many was to do it as you can imagine. For PDEs, there are lots of ways to slice and dice scalar fields and there are also lots of was to show vectors. It just depends on what you're looking for. There are techniques for everything. The answer to all of your questions is "yes".

What problem are you studying?

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Visualization is not a straightforward task and you can do countless things wrong when you try to visualize some complex result. Sometimes you may think you have a super smart and compact way to depict something but your readers will simply not understand what you mean. This refers mainly to artificially adding dimensions to plots, which I think should be generally avoided (though many exceptions exist). For instance, if you have a one-dimensional time-dependent PDE to solve and you plot it as an elevation plot in two dimensions this may seem appealing, since you can display the whole solution in one plot. However, if this is not a standard way to plot results in your community, it might be really confusing and in such cases it may be much more appropriate to show a series of plots at different times and let the user fill in the temporal gaps in his head. As a rule of thumb, I always try to adhere to what the visualization frameworks offer and try to be close to what others did before me, not because I believe it is THE best way, but because I know that readers are used to a certain way of presentation. I think of this in the same way as for graphical user interfaces. You can argue the whole day that combo boxes are a terrible way to make users choose among different options, or that the OK button should be on top rather than left of of the Cancel button in a message dialog, but it is difficult to convince everybody else to adhere to your way of thinking. Of course, when there is no standard way to represent data you can try to be be creative. But you will not often run into situations where you have to visualize something which nobody ever visualized before and it pays to make it easy for the reader to interpret your results. If you try to visualize in unexplored territories and have any doubt, ask the experts. There are visualization chairs at most universities and they are there for a reason.

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