Projection Methods for ODEs with Invariants

In this chapter we apply the two projection methods to ordinary differential equations with invariants, see Sec. 5.1.4. 

First we show that linear invariants are preserved by linear multistep methods, so that in that case no projections need to be applied. There will be no drift off from the manifold given by the invariant. 

Theorem 5.3.3 Assume that the differential equation x = f t,x) with x(0) = xq has a linear integral invariant Hx—b = 0. Given k starting values i = 0. k— 1 with Hxi -6 = 0, the solution x of the problem discretized by a consistent linear k step method satisfies this invariant toOy i.e. Hxn — 6 = 0. 

The proof is straightforward and uses the property Hf = 0, Def. 5.1.9, [Gear86]. This result no longer holds for linear invariants of the type 

We pointed out earlier that for equations of motion of constrained mechanical systems written in index-1 form the position and velocity constraints form integral invariants, see (5.1.16). Thus the coordinate projection and the implicit state space method introduced in the previous section can be viewed as numerical methods for ensuring that the numerical solution satisfies these invariants. 

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