Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Abstract

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

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

“…The semi-implicit CD method [12][13][14], which is a generalization of the Störmer-Verlet method for non-Hamiltonian systems, was chosen as a basic symmetric method for this study. We will show how its semi-explicit and semi-implicit variants can be used to construct an efficient error estimator.…”

Preference and Stability Regions for Semi-Implicit Composition Schemes

“…The semi-implicit integration CD method, which is a generalization of the Störmer-Verlet method over the arbitrary separable IVP, was previously described in [12] and is chosen as a basic method for the proposed composition schemes as it possesses high computational efficiency [21]. Due to its semi-implicit calculation nature, it exists only for ODE systems of order two and higher.…”

Stability Analysis and Optimization of Semi-Explicit Predictor–Corrector Methods

“…Since the investigated semi-explicit and semi-implicit multistep methods do not exist for first-order ODE systems, Dahlquist's test equation is not suitable for studying their stability. Several new approaches to study numerical stability have recently been proposed [11,[20][21][22]. Following the ideas described in [11,22], we chose the two-dimensional problem to plot stability regions.…”

“…Several new approaches to study numerical stability have recently been proposed [11,[20][21][22]. Following the ideas described in [11,22], we chose the two-dimensional problem to plot stability regions.…”

“…Regardless of the ODE system properties, e.g., stiffness, scale, delay, etc., one can often apply the same integration method. However, this approach can entail high computational costs in long-term simulations or multiparametric research of large-scale models [11]. One of the most common solutions here is linear multistep methods.…”