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I am actually confused about time stepping for PDEs. Before, I was distretizing time using backward Euler method for implicit formulation and I get a system of Algebraic equations to solve. Now during my research when I read papers, I find they disctretize space only leaving time derivative which yields system ODEs? What is the difference between two approaches and when I use one not another?

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If you just discretize space and turn your PDE system into a system of ODEs, you end up being able to take advantage of ODE integration libraries/codebases, which might make your life easier since you only spend the effort doing the spatial discretization and then get to just throw in those dynamics into a sort of black box ODE integrator.

If you discretize time and space, you really put the pressure on yourself to completely tackle and solve the problem on your own. This gives you more control over what time integration methods you want to implement and will likely make it easier to implement implicit schemes, for example. However, you do have more room for error and this approach can be more effort on paper and in implementation, depending on the scale of the problem.

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The technique of discretising the spatial derivatives in a problem and leaving the temporal evolution as continuous is known as the "method of lines", there's lots of stuff on google about it, e.g. http://www.scholarpedia.org/article/Method_of_lines. It's known as the method of lines because you literally solve the problem by solving individual equations (ODEs) along lines of constant $x$ (say).

One other possible advantage to using method of lines (besides those mentioned in the other answer) is avoiding the problem of consistency. When solving a PDE you can encounter the problem where your approximation of it is inconsistent with the PDE that you're trying to solve, and so you actually end up solving the wrong PDE. But by reducing the PDE to a system of ODEs you avoid the possibility of this happening (there is no equivalent of inconsistency when solving ODEs).

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