I have a code that uses both spatial and time discretization/integration. For convergence analysis, I am wondering how one would test the order of accuracy of their ${time}$ integration scheme? I assume you would have to isolate the time discretization from the spatial discretization somehow, but I am not sure how this can be done on a discretized mesh. I suppose you could choose some really really small spatial step size such that the spatial error is less than the time error, but this seems very inelegant and complicated.
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$\begingroup$ What are you considering as the truth in your analysis? $\endgroup$– origimboCommented Jun 7, 2018 at 22:12
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$\begingroup$ I'm taking "truth" to mean an 'exact' solution. If that's correct, then the truth is a manufactured solution (I'm using the method of manufactured solutions). $\endgroup$– user27504Commented Jun 8, 2018 at 0:51
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If you fix the spatial discretization, then you are solving a fixed system of ODEs. The exact solution of that system is not the same as the exact solution of the PDE, of course, but you can test convergence to it just like you would with any ODE system.
Since the exact solution is usually too complicated to work out by hand, one typically uses a highly refined (in time) solution as a substitute.
answered Jun 8, 2018 at 11:43
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$\begingroup$ I understand. Would this method work to determine the order of accuracy of my time integration method, by comparing my numerical solution to a hypothetical exact solution, where the hypothetical exact solution is just the solution obtained using some very small $\Delta t$? $\endgroup$ Commented Jun 8, 2018 at 17:25
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$\begingroup$ Yes, or more precisely it would work to verify the order of accuracy. $\endgroup$ Commented Jun 10, 2018 at 17:40