The ability to advance locally-stiff systems of equations in time depends on accurate and efficient temporal schemes. Recently, a new family of Paired Explicit Runge-Kutta (P-ERK) methods has been proposed. This approach allows different Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria. Whereas the original P-ERK formulation was only second-order accurate, in this paper we propose a new third-order family. We present the general formulation for these schemes, and then optimize them for the high-order discontinuous Galerkin method recovered via the flux reconstruction approach. We then verify that these schemes obtain their designed third-order accuracy for non-linear systems of equations. Performance results show that they achieve speedup factors up to four relative to a classical third-order Runge-Kutta methods for laminar and turbulent flow over an SD7003 airfoil, and turbulent flow over a tandem sphere configuration. Based on these results, this new family of third-order P-ERK schemes provides an appealing approach for accurate and efficient solution of locally-stiff systems of equations.

及时推进局部刚性方程组的能力取决于准确和有效的时间方案。最近，已经提出了一个新的成对显式龙格-库塔 (P-ERK) 方法家族。这种方法允许根据局部刚度标准分配具有不同活动级数的不同 Runge-Kutta 方案。虽然原始的 P-ERK 公式仅具有二阶精度，但在本文中，我们提出了一个新的三阶族。我们提出了这些方案的一般公式，然后针对通过通量重建方法恢复的高阶不连续 Galerkin 方法对其进行优化。然后，我们验证这些方案是否获得了它们为非线性方程组设计的三阶精度。性能结果表明，对于 SD7003 翼型上的层流和湍流以及串联球体配置上的湍流，它们相对于经典的三阶 Runge-Kutta 方法实现了高达 4 的加速因子。基于这些结果，这个新的三阶 P-ERK 方案家族为精确有效地求解局部刚性方程组提供了一种有吸引力的方法。