Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

A new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering (R-matrix) method. The method proposed allows the author to treat open quantum chains with appropriate boundary terms in the Hamiltonian. The general considerations are applied to the XXZ and XYZ models, the nonlinear Schrodinger equation and Toda chain.

https://math.berkeley.edu/~reshetik/Sklyanin-boundary.pdf

Boundary conditions for integrable quantum systems

This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.

/pdf/discrete-mechanics-and-variational-integrators-4qm2t5dk6u.pdf

Discrete mechanics and variational integrators

https://digital.csic.es/bitstream/10261/15682/1/blanes.pdf

The Magnus expansion and some of its applications

Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.

New similarity reductions of the Boussinesq equation

Given a quadratic vector field on \\mathbb{R}^n possessing a quadratic first integral depending on two of the independent variables, we give a constructive proof that Kahan's discretization method exactly preserves a nearby modifed integral. Building on this result, we present a family of integrable quadratic vector fields (including the Euler top) whose Kahan discretization is a novel 10-parameter family of integrable maps.

/pdf/geometric-and-integrability-properties-of-kahan-s-method-the-4ghwbv9u6s.pdf

Geometric and integrability properties of Kahan’s method: the preservation of certain quadratic integrals

A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. The proposed techniques has the property that all the time-steps are positive.

/pdf/symmetric-spaces-and-lie-triple-systems-in-numerical-th2rbkgroj.pdf

Symmetric spaces and Lie triple systems in numerical analysis of differential equations

Lax representation for integrable ODE's.

The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh–Abe methods. These are generalisations of the Itoh–Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh–Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency.

/pdf/a-geometric-integration-approach-to-nonsmooth-nonconvex-22dncjs4ob.pdf

A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation

We prove the Liouville and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization.

/pdf/liouville-integrability-and-superintegrability-of-a-2zjl0badnq.pdf

G. R. W. Quispel

Papers

Geometric and integrability properties of Kahan’s method: the preservation of certain quadratic integrals

Symmetric spaces and Lie triple systems in numerical analysis of differential equations

Lax representation for integrable ODE's.

A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation

Liouville integrability and superintegrability of a generalized Lotka-Volterra system and its Kahan discretization