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In this paper we present two semi-implicit-type second order CompactApproximate Taylor (CAT2) numerical schemes and blend them with a local aposteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solvehyperbolic systems of balance laws with relaxed source term. The resultingscheme presents high accuracy when applied to smooth solutions, essentiallynon-oscillatory behavior for irregular ones, and offers a nearly fail-safeproperty in terms of ensuring positivity. The numerical results obtained from avariety of test cases, including smooth and non-smooth well-prepared andunprepared initial condition, assessing the appropriate behavior of thesemi-implicit-type second order CATMOOD schemes. These results have beencompared in accuracy and efficiency with a second order semi-implicitRunge-Kutta (RK) method.

Semi-Implicit-type order-adaptive CAT2 schemes for systems of balance laws with relaxed source term