Abstract In the past years several unconditionally positivity preserving numerical methods for production-destruction equations arising in a wide variety of real life applications were developed. The BBKS schemes suggested by Bruggemann et al. (2007) [4] and Broekhuizen et al. (2008) [3] modifications of Heun's method and conserve all linear invariants while still maintaining positivity independent of the time step size used. In this paper, we present a comprehensive generalization of these schemes as modifications of arbitrary first and second order Runge-Kutta methods with nonnegative parameters. This is achieved by the introduction of novel Patankar weights and sufficient and necessary conditions w.r.t. the Patankar weight denominators for first as well as second order consistency are proven. Furthermore, a convergence proof for the incorporated Newton scheme is given such that the efficiency compared to the original BBKS schemes, which make use of the bisection method is significantly improved. Numerical simulations are presented confirming the theoretical results concerning the validity, efficiency and accuracy of the new class of generalized BBKS schemes.

中文翻译：

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广义BBKS方案的综合理论

摘要 在过去的几年中，开发了几种用于生产-破坏方程的无条件正性保持数值方法，这些方法出现在广泛的现实生活应用中。Bruggemann 等人提出的 BBKS 方案。(2007) [4] 和 Broekhuizen 等人。(2008) [3] 修改 Heun 的方法并保留所有线性不变量，同时仍然保持独立于所使用的时间步长的正性。在本文中，我们将这些方案作为对具有非负参数的任意一阶和二阶 Runge-Kutta 方法的修改进行全面概括。这是通过引入新的 Patankar 权重和充分必要条件来实现的，并且证明了 Patankar 权重分母的一阶和二阶一致性。此外，给出了合并牛顿方案的收敛性证明，与使用二分法的原始 BBKS 方案相比，效率得到显着提高。数值模拟证实了关于新型广义 BBKS 方案的有效性、效率和准确性的理论结果。