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.

翻译：本文提出两种半隐式型二阶紧致近似泰勒（Compact Approximate Taylor, CAT2）数值格式，并将其与局部后验多维最优阶检测（Multi-dimensional Optimal Order Detection, MOOD）范式结合，用于求解带松弛源项的双曲型平衡律系统。该格式在求解光滑解时具有高精度，对非规则解呈现本质无振荡行为，并在确保正值性方面提供近乎万无一失的特性。通过一系列测试算例（包括光滑与非光滑的良好预备及非预备初始条件）获得的数值结果，验证了半隐式型二阶CATMOOD格式的恰当性能。这些结果在精度与效率上已与二阶半隐式龙格-库塔（Runge-Kutta, RK）方法进行了比较。