Lie–Poisson Methods for Isospectral Flows

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-019-09428-w Lie–Poisson Methods for Isospectral Flows Klas Modin1 · Milo Viviani1 Received: 8 August 2018 / Revised: 8 May 2019 / Accepted: 11 May 2019 © The Author(s) 2019 Abstract The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie–Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectrum in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail to preserve the underlying Lie–Poisson structure. Here, we present a class of numerical methods of arbitrary order for Hamiltonian and non-Hamiltonian isospectral flows, which preserve both the spectra and the Lie–Poisson structure. The methods are surprisingly simple and avoid the use of constraints or exponential maps. Furthermore, due to preservation of the Lie–Poisson structure, they exhibit near conservation of the Hamiltonian function. As an illustration, we apply the methods to several classical isospectral flows. Keywords Isospectral flow · Lie–Poisson integrator · Symplectic Runge–Kutta methods · Toda flow · Generalized rigid body · Chu’s flow · Bloch–Iserles flow · Euler equations · Point vortices Mathematics Subject Classification 37M15 · 65P10 · 37J15 · 53D20 · 70H06 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Reduction Theory for Isospectral Lie–Poisson Integrators . . . . . . . . . . . . . . . . . . . . . . . 4 Isospectral Symplectic Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Communicated by Arieh Iserles. B Klas Modin Milo Viviani 1 Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Gothenburg, Sweden 123 Foundations of Computational Mathematics 4.1 Symplectic Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Partitioned Symplectic Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Linear Equivariance of the Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Generalized Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The (Periodic) Toda Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Euler Equations on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Point Vortices on a Sphere and the Heisenberg Spin Chain . . . . . . . . . . . . . . . . . . . . . 5.5 The Bloch–Iserles Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The Toeplitz Inverse Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 The Brockett Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Lie–Poisson systems and isospectral flows are two well-studied classes of dynamical systems. The former appear as Poisson reductions of Hamiltonian systems for which the configuration and symmetry space is a Lie group (see the monograph [17] and references therein). The classical example is the free rigid body as viewed by Poincaré [26]. The latter, isospectral flows, appear as Lax formulations of integrable systems (see the survey papers [5,30,31] and references therein). The classical example is the Toda lattice as viewed by Flaschka [8,29]. The study of numerical methods for the two classes of systems is by now classical subjects in numerical analysis. The motivation for such schemes came through the strong connection between matrix factorizations in numerical linear algebra and isospectral flows (see the survey papers [6,23]). This was initiated by the remarkable discovery that the iterative Q R-algorithm for computing eigenvalues is a discretization of the (non-periodic) Toda flow [7,28]. The general form of an isospectral flow is Ẇ = [B(W ), W ], W ∈ S ⊂ gl(n, C). (1) Here, [·, ·] denotes the matrix commutator, S is a linear subspace of the Lie algebra gl(n, C), and the function B : S → n(S) maps into the normalizer algebra n(S) (see Sect. 3 for details). The most studied setting is when S = Sym(n, R) is the space of symmetric real matrices, for which the normalizer is the Lie algebra of skewsymmetric real matrices n(S) = so(n). Another setting is when S = g is a Lie subgroup of gl(n, C), for which the normalizer is the subalgebra itself n(S) = g. Let us now discuss the connection between isospectral flows and Lie–Poisson systems. The predominant example connecting the two is Manakov’s n-dimensional rigid body [16]. Recall that a Lie–Poisson system evolves on the dual g∗ of a Lie algebra g. Given a Hamiltonian function H on g∗ , the flow W (t) ∈ g∗ is given by Ẇ = ad∗d H (W ) (W ), 123 (2) Foundations of Computational Mathematics where the operator ad∗ is defined by ∗ (W ), V  = W , [U , V ] adU ∀ U , V ∈ g. (3) Without loss of generality, we may assume that g is a subalgebra of gl(n, C). To identify gl(n, C)∗ with gl(n, C), we use the Frobenius inner product W , V  = Tr(W † V ), where W † denotes the conjugate transpose. In this way, we also identify g∗ with the subspace g ⊂ gl(n, C). Next we extend the Hamiltonian to all of gl(n, C) by taking it to be constant on the affine spaces given by translations of the orthogonal complement of g. Then, d H corresponds to ∇ H . From definition (3) and the identification of g∗ with g, we get ad∗W (M) =  [W † , M], where  is the orthogonal projection gl(n, C) → g. We thus arrive at an explicit formulation of the Lie–Poisson system (2), namely Ẇ =  [∇ H (W )† , W ]. (4) Now, the key observation is that if the representation of g as a subalgebra of gl(n, C) is closed under conjugate transpose, then Eq. (4) becomes the isospectral flow Ẇ = [∇ H (W )† , W ]. (5) Such a representation is possible if and only if g is a reductive Lie algebra (see Sects. 2–3 for details). Thus, we arrive at the statement that Lie–Poisson systems for any reductive Lie algebra can be viewed as isospectral flows. Recall that most classical Lie algebras are reductive, for example gl(n, C), gl(n, R), sl(n, C), sl(n, R), u(n), su(n), so(n), and sp(n). An interesting consequence of Eq. (5) is that whenever the function B(W ) in the isospectral flow (1) can be written as B(W ) = ∇ H (W )† , then it can be extended to a Lie–Poisson system on gl(n, C) (or possibly a smaller reductive Lie algebra contai (...truncated)