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

Eigenspectra of the critical quantum Ashkin-Teller and Potts chains with free boundaries can be obtained from that of the XXZ chain with free boundaries and a complex surface field. By deriving and solving numerically the Bethe ansatz equations for such boundaries the authors obtain eigenenergies of XXZ chains of up to 512 sites. The conformal anomaly and surface exponents of the quantum XXZ, Ashkin-Teller, and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.

Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models

This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs). Given a...

Geometric integration using discrete gradients

Integrable mappings and soliton equations II

The first ever energy-preserving B-series numerical integration method for (ordinary) differential equations is presented and applied to several Hamiltonian systems. Related novel Lie algebraic results are also discussed.

https://iopscience.iop.org/article/10.1088/1751-8113/41/4/045206/pdf

G. R. W. Quispel

Papers

Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models

Geometric integration using discrete gradients

Integrable mappings and soliton equations II

A new class of energy-preserving numerical integration methods

Integrable mappings and soliton equations