New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers

Abstract

Composition is a powerful and simple approach for obtaining numerical integration methods of high accuracy order while preserving the geometric properties of a basic integrator. Adaptive step size control allows one to significantly increase the performance of numerical integration methods. However, there is a lack of efficient step size control algorithms for composition solvers due to some known difficulties in constructing a low-cost embedded local error estimator. In this paper, we propose a novel local error estimator based on a difference between the semi-implicit CD method and semi-explicit midpoint methods within a common composition scheme. We evaluate the performance of adaptive composition schemes with the proposed local error estimator, comparing it with the other state-of-the-art approaches. We show that composition ODE solvers with the proposed step size control algorithm possess higher numerical efficiency than known methods, by using a comprehensive set of nonlinear test problems.

Comparison of the Performance of New and Traditional Numerical Methods for Long-Term Simulations of Heat Transfer in Walls with Thermal Bridges

“…It has been shown that explicit approaches can be more effective even when tiny time step sizes are required [20]. Moreover, various ingenious combinations of explicit and implicit methods, such as semi-explicit or semi-implicit methods, have also been presented [21][22][23][24]. However, they do not really solve the problems and the drawbacks of the explicit and implicit approaches discussed above.…”

Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates

“…Beuken et al [24] constructed versions of these methods based on the backward differentiation formula and Adams-Bashforth schemes for ODE systems, which can be an order of magnitude faster than some traditional methods. Fedoseev et al [25] improved the performance of semi-implicit composition integration methods by variable step size control algorithms. The paper of Ji and Xing [26] proposes a set of time-marching methods which uses the generalized Padé approximation, the Gauss-Legendre quadrature, and explicit Runge-Kutta schemes to solve systems of ODEs.…”