Quasi-linear hyperbolic systems with source terms introduce significant computational challenges due to the presence of a stiff source term. To address this, a finite volume Nessyahu-Tadmor (NT) central numerical scheme is explored and applied to benchmark models such as the Jin-Xin relaxation model, the shallow-water model, the Broadwell model, the Euler equations with heat transfer, and the Euler system with stiff friction to assess their effectiveness. The core part of this numerical scheme lies in developing a new implicit-explicit (IMEX) scheme, where the stiff source term is handled in an semi-implicit manner constructed by combining the midpoint rule in space, the trapezoidal rule in time with a backward semi-implicit Taylor expansion. The advantage of the proposed method lies in its stability region and maintains robustness near stiffness and discontinuities, while asymptotically preserving second-order accuracy. Theoretical analysis and numerical validation confirm the stability and accuracy of the method, highlighting its potential for efficiently solving the stiff hyperbolic systems of balance laws.

翻译：具有源项的拟线性双曲系统因存在刚性源项而带来显著的计算挑战。为应对此问题，本文探索并应用了有限体积Nessyahu-Tadmor（NT）中心数值格式，将其应用于基准模型进行评估，包括Jin-Xin松弛模型、浅水模型、Broadwell模型、带热传导的欧拉方程以及带刚性摩擦的欧拉系统，以检验其有效性。该数值格式的核心在于发展了一种新的隐式-显式（IMEX）格式，其中刚性源项以半隐式方式处理，其构造结合了空间中的中点法则、时间上的梯形法则以及后向半隐式泰勒展开。所提方法的优势在于其稳定区域，并在刚性和间断附近保持鲁棒性，同时渐近地保持二阶精度。理论分析和数值验证证实了该方法的稳定性与精度，突显了其在高效求解刚性守恒律双曲系统方面的潜力。