The study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system, whose discretisation is related to reverse accumulation in automatic differentiation. Similar adjoint systems arise in optimal control and other areas, including classical Mechanics. Adjoint systems are introduced in...

The non-linear wave equation is taken as a model problem for the investigation. Different multisymplectic reformulations of the equation are discussed. Multi-symplectic Runge–Kutta methods and multi-symplectic partitioned Runge–Kutta methods are explored based on these different reformulations. Some popular and efficient multi-symplectic schemes are collected and constructed. Stability analyses...

We present new symmetric fourth and sixth-order symplectic Partitioned Runge{ Kutta and Runge{Kutta{Nystrr om methods. We studied compositions using several extra stages, optimising the eeciency. An eeective error, E f , is deened and an extensive search is carried out using the extra parameters. The new methods have smaller values of E f than other methods found in the literature. When applied...

We consider canonical partitioned Runge-Kutta methods for separable Hamiltonians H = T(ß) + Viq) and canonical Runge-Kutta-Nyström methods for Hamiltonians of the form H = ^pTM~lp + Viq) with M a diagonal matrix. We show that for explicit methods there is great simplification in their structure. Canonical methods of orders one through four are constructed. Numerical experiments indicate the sui...

Separable Hamiltonian systems of differential equations have the form dp/dt = -dH/dq, dq/dt = dH/dp, with a Hamiltonian function H that satisfies H = T(p) + K(q) (T and V are respectively the kinetic and potential energies). We study the integration of these systems by means of partitioned Runge-Kutta methods, i.e., by means of methods where different Runge-Kutta tableaux are used for the p and...

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total...

This work generalizes the additively partitioned Runge-Kutta methods by allowing for different stage values as arguments of different components of the right hand side. An order conditions theory is developed for the new family of generalized additive methods, and stability and monotonicity investigations are carried out. The paper discusses the construction and properties of implicit-explicit ...