I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values which smoothly extend the functions $u^+$ and $u^-$ from either side of the interface. These fictitious values are second order. I've implemented this numerically and have second order convergence in space but only first order in time. Crank-Nicolson is second order in time so my results perplex me. I've tried several different interfaces and they all give me first order convergence in time. The interface is not moving so I have no discontinuities between time steps for my time derivative. Any suggestions would be appreciated.
2026-03-26 04:29:32.1774499372
heat equation with Interface Crank Nicolson
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