On conjugate difference schemes: the midpoint scheme and the trapezoidal scheme

The preservation of quadratic integrals on approximate solutions of autonomous systems of ordinary differential equations x ˙ = f ( x ) , found by the trapezoidal scheme, is investigated. For this purpose, a relation has been established between the trapezoidal scheme and the midpoint scheme, which preserves all quadratic integrals of motion by virtue of Cooper’s theorem. This relation allows considering the trapezoidal scheme as dual to the midpoint scheme and to find a dual analogue for Cooper’s theorem by analogy with the duality principle in projective geometry. It is proved that on the approximate solution found by the trapezoidal scheme, not the quadratic integral itself is preserved, but a more complicated expression, which turns into an integral in the limit as ∆ t → 0 . Thus the concept of conjugate difference schemes is investigated in pure algebraic way. The results are illustrated by examples of linear and elliptic oscillators. In both cases, expressions preserved by the trapezoidal scheme are presented explicitly.


Introduction
Dynamical systems are the most important mathematical models in mechanics and physics. Only a few of these models are integrated in a closed form [1], therefore, they have to be investigated using numerical methods, of which the most important is the finite difference method.
Let be a point in an -dimensional affine space. Any difference scheme that approximates differential equation describes a transition from the value at some initial moment of time to the value of̂at the moment of time shifted from the initial value by the quantity Δ , called the step. We will consider algebraic schemes, i.e., those in which the above correspondence is specified using a system of algebraic equations ( ,, Δ ) = 0.
If the original equation has an algebraic integral ( ) = , and it follows from the equations (2) that () = ( ), then this difference scheme is said to preserve this integral.
If we use explicit difference schemes for integrating dynamical systems, then the values of the integrals of motion will change monotonically step by step. At the turn of the 1980s and 1990s, the first difference schemes were constructed that preserve exactly the algebraic integrals of dynamical systems. For example, the scheme constructed by D. Greenspan preserves all classical integrals of N-body problems [2]- [5], the symplectic Runge-Kutta schemes, including the simplest of them, the midpoint schemê preserve linear and quadratic integrals in virtue Cooper's theorem [6]- [9]. This circuit has a whole bunch of wonderful properties inherited from the original differential equation [10]. This seems to be a simple consequence of the -symmetry of the midpoint circuit: the equation (3) is invariant under the transformation The trapezoidal scheme has the same propertŷ however, in experiments with an elliptic oscillator performed by Yu. A. Blinkov for PCA'2019 [11], the quadratic integrals oscillated, although they did not increase monotonically. The absence of monotonicity in the variation of the values of the integrals of motion on approximate solutions is extremely important from the physical point of view, since, on average, all fundamental conservation laws are satisfied on solutions of this type. The noted behavior of the approximate solutions found by the trapezoidal scheme can be explained by the fact that it is conjugated to the midpoint scheme and therefore some more complex expression is retained on it [9, §VI.8.1 -2].
The very concept of conjugate difference schemes [9, def 8.1] is formulated locally in terms of power series. The implicit function theorem can be applied to the system of algebraic equations (2) and, under certain conditions, we can assert that where Φ is a series in powers of Δ , the coefficients of which are rational functions of . Difference schemeŝ are referred to as mutually conjugate, if there exists a change of coordinates It is clear from this definition that the exact preservation of the expression of one of these schemes entails the preservation of some expression by the other scheme. Say, if the scheme Ψ Δ preserves the integral ( ) exactly, then the scheme Φ Δ preserves the expression In this article, we will clarify the geometric meaning of the conjugacy of the two above schemes and write down explicitly the expression that preserves the trapezoidal scheme.

Relationship between trapezoidal and midpoint schemes
The approximate solution of the system (1), found by the scheme (2) with constant step Δ , is a finite or infinite sequence of points the first element of which is chosen in an arbitrary way, and all the others are defined recursively: +1 is the root̂of the equation Proof. The middle of link +1 of the solution found by the midpoint scheme is given by the formula In this case, the ends of the link are unambiguously reconstructed from its given midpoint: Since +1 belongs to two links +1 and +1 +2 , we have Thus, the midpoints can be calculated using the trapezoidal difference scheme (4).
By virtue of the theorem 1, the solutions 0 1 2 … and ′ 0 ′ 1 ′ 2 …, found by the midpoint schemes (3) and by the trapezoidal scheme (4) turn out to be coupled with each other. By analogy with the duality principle in projective geometry [12] it is hoped that any statement about the midpoint scheme should have a 'twin' in the trapezoidal scheme.
One of the most interesting properties of the midpoint scheme is Cooper's theorem [9, th. 2.2], according to which this scheme preserves any quadratic integral of motion.
is easily rewritten by expressing though ′ , and +1 though ′ +1 : Therefore, for dual scheme (4) the conservation law takes the form Thus, a quadratic integral is also inherited by the trapezoidal scheme (4), but the expression for the conserved quantity coincides with only in the limit Δ → 0. This circumstance made complicated finding it.

Definition 1.
Let us say that a difference scheme inherits the integral ( ) = if there exists a rational function ( , Δ ) such that 1) from the equations that specify the scheme it follows that (, Δ ) = ( , Δ ), 2) in the limit Δ → 0 expression ( , Δ ) turns into ( ) Function itself will be referred to as the difference analog of the integral .

Examples
Consider several examples.

Linear oscillator
In the case of a linear dynamical system, the midpoint scheme and the trapezoidal scheme are the same, so the midpoint scheme becomes selfconjugate. This circumstance greatly simplifies the study of the midpoint scheme for a linear oscillator.
Consider a dynamic systeṁ= − ,̇= which has a quadratic integral The midpoint scheme will give points ( 0 , 0 ), ( 1 , 1 ), … lying on the circle the radius of which is determined by the initial point The midpoints of the links lying on the circle the radius of which can be determined by the first link Thus, the trajectory on the phase plane turns out to be a broken line, the vertices of which lie on a circle of radius , and the links touch a concentric circle, the radius of which is √ 1 + Δ 2 /4 times less than . In particular, the trajectory will be closed, and the solution will be periodic if and are the radii of the circumscribed and inscribed circle in the -gon, that is, if This immediately gives the formula for choosing a step This formula was previously obtained by us analytically [10].

Elliptic oscillator
By the definition of Jacobi functions [13], is a particular solution of the autonomous system of differential equationṡ with the initial conditions = 0, = = 1 at = 0.
The midpoint scheme preserves both intergals 2 + 2 = const and 2 2 + 2 = const (9) of this system. Now the trapezoidal schemê does not coincide with the midpoint scheme and, therefore, its invariants are more complicated. Nevertheless, the integral 2 + 2 = const corresponds to the integral ) .
The integral 2 2 + 2 = const correspods to Thus, in the space the vertices of the trajectory lie on the elliptic curve (9), and the midpoints of the links of the broken line lie on a more complex curve This means that the trapezoidal scheme for an elliptic oscillator inherits both quadratic integrals, and their difference counterparts are the expressions (10).
If we follow the change in 2 + 2 on the approximate solution found by the trapezoidal scheme, then we will see a deviation from a constant value equal to The exact solution is periodic, so in the plots these deviations appear as periodic fluctuations.

Conclusion
If you do not use specially developed difference schemes, discretization of continuous models by the method of finite differences introduces completely new properties into these models: calculations lead to a monotonic change in quantities, which, from physical considerations, must remain constant. For example, in computer experiments, dissipation appears even in those cases when energy was conserved in the original continuous model. In calculations for sufficiently long time intervals, this dissipation becomes very noticeable, and the parameters of the dynamical system are significantly distorted.
The number of schemes that preserve algebraic integrals of motion exactly is small and their drawbacks are well known. Difference schemes, in which the integrals of motion fluctuate around their initial values, significantly expand this set. However, the noted property is usually accepted without explanation and even more rigorous proof. Theorem 2, which is dual to Cooper's theorem, allows us to fill in this gap for the trapezoidal scheme (4) by explicitly specifying expressions that coincide in the limit Δ → 0 with exact integrals and at the same time are preserved on approximate solutions exactly.