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I'm looking at page 10 of http://www4.ncsu.edu/~zhilin/TEACHING/MA402/notes1.pdf

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What happened to the boxed term (it doesn't seem to appear in the matrix equation).

(I'm just implementing this and am not familiar with pde/numerical techniques)

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Looks like an error in the notes. You do need the boxed term to appear in the equation; each component of the vector on the right-hand side should have an additional $\frac1{\Delta t}u_i^k$ term.

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