= Stability of a two-parameter implicit-explicit diffusion scheme
For the scheme
$$
\left(aI-\frac{\mu-c}{4}L\right)u^{n+1}
=\left(aI+\frac{\mu+c}{4}L\right)u^n,
$$
where $L=[1,-2,1]$ is the Dirichlet second-difference matrix and $\mu>0$, the modal amplification factor is
$$
G(s)=\frac{a-(\mu+c)s}{a+(\mu-c)s},
\qquad 0<s<1.
$$
Mesh-uniform stability holds exactly when
$$
a\geq0,
\qquad
c\leq a.
$$
For consistency with $u_t=u_{xx}$ under the displayed normalization one additionally chooses $a=1/2$; stability then requires $c\leq1/2$.
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