Stability of a two-parameter implicit-explicit diffusion scheme

For the schemewhere $L=[1,-2,1]$ is the Dirichlet second-difference matrix and $\mu >0$, the modal amplification factor isMesh-uniform stability holds exactly whenFor consistency with $u_t=u_{xx}$ under the displayed normalization one additionally chooses $a=1/2$; stability then requires $c\le 1/2$.