Semi-Implicit Central scheme for Hyperbolic Systems of Balance Laws with Relaxed Source Term

Abstract

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 a 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. The numerical validation further extends to two-dimensional configurations of the Jin-Xin relaxation model and a Jin-Xin-type relaxation system of 2D Euler equation. 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 of 1D and 2D.