This study presents a systematic performance evaluation of three flux‑reconstruction strategies in the discrete unified gas‑kinetic scheme (DUGKS) for low‑speed continuum flows. The collision term integral at cell interfaces is approximated with the left‑rectangular, trapezoidal (standard DUGKS) and right‑rectangular rules, while a unified finite‑volume framework is retained elsewhere.  Theoretical analysis confirms that all schemes retain second‑order accuracy; however, numerical benchmarks (Taylor–Green vortex, lid‑driven cavity, and flow past a square cylinder) reveal clear performance differences. On the same meshes, the right‑rectangular and trapezoidal rules produce nearly equal errors, with the right‑rectangular rule marginally superior, whereas the left‑rectangular rule is an order of magnitude less accurate. Per time step, the rectangular rules are faster than the trapezoidal rule; when total runtime to convergence is considered, the right‑rectangular rule is the most economical and markedly more stable, remaining robust for exceptionally large ratios between the time step and the relaxation time. Hence, the right‑rectangular formulation offers the best overall balance of accuracy, efficiency, and stability for continuum flow simulations.

discrete unified gas-kinetic scheme,integration scheme,low-speed continuum flow