Courant-Like Conditions Limit Reasonable Mesh Refinement to Order $h^2 $

Under conditions (e.g., Courant-Friedrichs-Lewy conditions) on the size of all local time steps—namely, that they be proportional to the local spatial mesh size–a mesh refinement algorithm cannot be a “reasonable” algorithm if, near some space-time surface, the spatial mesh is refined to size $o(h^2 )$, i.e., smaller than order $h^2 $. Here h typifies the mesh size over most of the spatial region; and an algorithm is called “unreasonable” if, asymptotically, it spends all of its computational work exercising at the space-time surface. The limitation is even more stringent for problems with more than one spatial dimension unless the refinement occurs only normal to the surface. As the local time step constraints may be imposed for reasons of accuracy or efficiency as well as for stability, they may differ, on the coarse or fine meshes, from Courant-like conditions. In that case, the limitations on the order of reasonable refinement will generally change. But there is an absolute limitation which depends on the time step restriction away from the surface and on the number of spatial dimensions. One consequence of all this appears to be that $o(h^4 )$ errors are unattainable in vast regions of space-time utilizing reasonable mesh refinement–at least when using more or less uniformly applied difference schemes, with Courant-like conditions, to approximate those solutions of hyperbolic systems whose first derivatives or values jump at some surface in space-time.