# Numerical Integration Schemes Based on Composition of Adjoint Multistep Methods
## 1 Introduction  
Ordinary differential equations (ODEs) are one of the most common mathematical models describing physical, biological, social, technical, and many other processes. A common tool for solving ODEs in digital computers is numerical integration methods. One of the most known examples of such algorithms is the well-known Runge-Kutta methods and linear multistep methods, which combine high computational efficiency and sufficient numerical stability. However, for some specific problems other properties of numerical integration methods are of great importance. Over the last 40 years, a new branch of numerical integration called geometric integration [1] has arisen. Geometric integration methods allow preserving the geometric properties of the original dynamical system like symplecticity, energy, or other first integrals. They also may possess symmetry, which helps to perform the simulation of physical systems more accurately and introduce the time-reversibility of the solution. Being initially applied mainly to Hamiltonian problems, geometric integration also showed its efficiency in solving non-Hamiltonian problems due to a smaller error constant in comparison with methods of the same order. Several symplectic modifications of Runge-Kutta [2; 3; 4] and linear multistep [5; 6; 7] methods were discovered recently. Two powerful approaches which are common in geometric integration are composition and splitting methods [8]. [PERSON] [9] and [PERSON] [10] developed a theoretical basis for a family of composition schemes that can be applied to an arbitrary symmetric basic method to increase its order of accuracy. [PERSON] et al. proposed several efficient composition and splitting schemes [11; 12; 13]. Recently some generalizations of thesymplectic Euler method [14] over a broader class of ODEs were proposed, allowing the application of composition methods to some non-Hamiltonian problems.  
Besides, in the case of the multistep methods, which are known for their low computational costs, the composition allows the breaking of the first and second Dahlquist barriers and obtaining strongly stable or even A-stable schemes. Examples of this sort are cyclic methods [15; 16; 17; 18] in which multiple multistep methods are applied periodically. The multistep block methods [19] are another perfect example of such integrators.  
It is known, that the composition of adjoint methods allows obtaining geometrical methods with higher accuracy order. The goal of the current study is to develop new numerical schemes by applying the composition technique to the linear multistep method and its adjoint, and the investigation of new integration methods in terms of key geometric properties and computational efficiency.  
The rest of the paper is organized as follows. In Section 2, a brief introduction to the numerical integration of ODE, multistep methods and composition principle are given. In Section 3, a theory for the construction of numerical schemes by the composition of the multistep method and its adjoint is reported. In Section 4, two new composition schemes are proposed and some of their properties are described. Section 5 reports the results of a practical investigation of the properties of designed numerical composition schemes. Finally, Section 6 concludes the paper and discusses the obtained results.