Lagrangian flow for fixed boundary problem

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Is it possible to use Lagrangian flow representation to solve a fixed-boundary problem? I have one-dimensional problem with a fixed boundary, and Lagrangian representation seems to be able to deal with shocks (that do occur in the problem) nicely by clustering particles in their vicinity; however I can't find any information on whether it is possible to use this particle-tracking approach when the outer boundary is set.