  In this paper we present two semi-implicit-type second order Compact
Approximate Taylor (CAT2) numerical schemes and blend them with a local a
posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve
hyperbolic systems of balance laws with relaxed source term. The resulting
scheme presents high accuracy when applied to smooth solutions, essentially
non-oscillatory behavior for irregular ones, and offers a nearly fail-safe
property in terms of ensuring positivity. The numerical results obtained from a
variety of test cases, including smooth and non-smooth well-prepared and
unprepared initial condition, assessing the appropriate behavior of the
semi-implicit-type second order CATMOOD schemes. These results have been
compared in accuracy and efficiency with a second order semi-implicit
Runge-Kutta (RK) method.
