Semi-Implicit Numerical Integration of Boundary Value Problems

The numerical solution to boundary differential problems is a crucial task in modern applied mathematics. Usually, implicit integration methods are applied to solve this class of problems due to their high numerical stability and convergence. The known shortcoming of implicit algorithms is high computational costs, which can become unacceptable in the case of numerous right-hand side function calls, which are typical when solving boundary problems via the shooting method. Meanwhile, recently semi-implicit numerical integrators have gained major interest from scholars, providing an efficient trade-off between computational costs, stability, and precision. However, the application of semi-implicit methods to solving boundary problems has not been investigated in detail. In this paper, we aim to fill this gap by constructing a semi-implicit boundary problem solver and comparing the performance of explicit, semi-implicit, semi-explicit, and implicit methods using a set of linear and nonlinear test boundary problems. The novel blinking solver concept is introduced to overcome the main shortcoming of the semi-implicit schemes, namely, the low convergence on exponential solutions. The numerical stability of the blinking semi-implicit solver is investigated and compared with existing methods by plotting the stability regions. The performance plots for investigated methods are obtained as a dependence between global truncation error and estimated computation time. The experimental results confirm the assumption that semi-implicit numerical methods can significantly outperform both explicit and implicit solvers while solving boundary problems, especially in the proposed blinking modification. The results of this study can be efficiently used to create software for solving boundary problems, including partial derivative equations. Constructing semi-implicit numerical methods of higher-accuracy orders is also of interest for further research.