<abstract><p>This paper proposes a new very simply explicitly invertible function to approximate the standard normal cumulative distribution function (CDF). The new function was fit to the standard normal CDF using both MATLAB's Global Optimization Toolbox and the BARON software package. The results of three separate fits are presented in this paper. Each fit was performed across the range $ 0 \leq z \leq 7 $ and achieved a maximum absolute error (MAE) superior to the best MAE reported for previously published very simply explicitly invertible approximations of the standard normal CDF. The best MAE reported from this study is 2.73e–05, which is nearly a factor of five better than the best MAE reported for other published very simply explicitly invertible approximations.</p></abstract>

Many mathematical models of natural phenomena are described by partial differential equations (PDEs) that consist of additive contributions from different physical processes. Classical methods for the numerical solution to such equations are monolithic in that all processes are treated with a single method. Additive numerical methods, in contrast, apply distinct methods to each additive term. There are, however, different ways mathematically to specify the additive terms, and it is not always clear which ways (if any) offer advantages over monolithic methods. This study compares the performance of two different additive splitting techniques (physics-based splitting and dynamic linearization) on a suite of eight test problems that involve advection, reaction, and diffusion with various 2-additive Runge–Kutta methods and (monolithic) Runge–Kutta–Chebyshev (RKC) methods. Results show that dynamic linearization generally outperforms physics-based splitting and so should be preferred as the splitting technique when splitting is required or otherwise desirable. RKC methods are the best performers on three of the eight problems, especially at coarse tolerances, but they can also be prone to severe underperformance.