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I've used an explicit finite difference scheme to model the 1D time dependent temperature distribution in a friction weld. I want to now verify the consistency and convergence of my algorithm.

I have no exact solution and assume I must use an approximation produced at a very refined mesh spacing to calculate the relative error for the coarser meshes. I plan to calculate the $L^2$ and $L^\infty$ error norms at a number of time intervals (during both heating and cooling). I am under the impression that the $L^2$ norm will provide the best overall description of the error, whereas the $L^\infty$ will allow me to bound my error.

So, is this is a logical and robust method by which to estimate my error and confirm the consistency of my difference scheme?

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Both the $L^2$ and the $L^\infty$ norm may be viable measures of the error. And if your problem is well natured, they might even behave the same way as your mesh spacing tends to zero. But if the solution to your problem is not smooth (think of a step function, which cannot be approximated by smooth functions in $L^\infty$), or your method has worse convergence in $L^\infty$, they may converge with different orders, if at all. If you observe that, you will have to look at the mathematics behind your method to see if your implementation is consistent.

By the way, you can simplify your error estimation: if you compute solutions on a sequence of meshes with spacing for instance $h_k = 2^{-k}$, and the difference of two consecutive solutions behaves like $h_k^\alpha$ with positive $\alpha$, you can use the gemoetric series to deduce that your method is of order $h^\alpha$. Comparinng solutions on consecutive meshes is usually easier.

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  • $\begingroup$ Are the potential problems with $L^2$ and $L^\infty$ due to the fact that my model is non-linear? I read a report by Strang, which suggested that if the solution has sufficient continuous derivatives then I should be able to analyze the stability in the same manner as I would if it were a linear problem. Does the same apply to consistency and convergence? $\endgroup$ – casimp Sep 18 '13 at 16:40
  • $\begingroup$ Thanks for the tip on using the geometric series - this does seem like a quicker method by which to look at the error! I was also wondering whether it was sensical to interrogate the solution at a number of time steps or am I just needlessly duplicating my effort? $\endgroup$ – casimp Sep 18 '13 at 16:51
  • $\begingroup$ Nonlinearity is not an issue. An example that comes to mind is the $L^\infty$ norm of the error of linear and bilinear finite elements, where you loose a logarithm on irregular meshes $\endgroup$ – Guido Kanschat Sep 19 '13 at 23:18

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