Although Runge–Kutta and partitioned Runge–Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrödinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on t...

Practical, structure-preserving methods for integrating classical Heisenberg spin systems are discussed. Two new integrators are derived and compared, including (1) a symmetric energy and spin-length preserving integrator based on a Red-Black splitting of the spin sites combined with a staggered timestepping scheme and (2) a (Lie-Poisson) symplectic integrator based on Hamiltonian splitting. Th...

We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combines adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge-Kutta methods in time. Weak formulation is only created for the stationary residual of the equation, and the Runge-Kutta method is supplied via its Butcher’s table. Ar...

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct th...

Maŕıa López-Fernández1, Christian Lubich2, Cesar Palencia1, and Achim Schädle3 1 Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain. E-mail: {marial, palencia}@mac.cie.uva.es 2 Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D–72076 Tübingen, Germany. E-mail: [email protected] 3 ZIB Berlin, Takustr. 7, D-14195 Berlin,...

We study numerical integrators that contract phase space volume even when the ODE does so at an arbitrarily small rate. This is done by a splitting into two-dimensional contractive systems. We prove a sufficient condition for Runge-Kutta methods to have the appropriate contraction property for these two-dimensional systems; the midpoint rule is an example.

In this paper a third-order composite Runge Kutta method is applied for solving fuzzy differential equations based on generalized Hukuhara differentiability. This study intends to explore the explicit methods which can be improved and modified to solve fuzzy differential equations. Some definitions and theorem are reviewed as a basis in solving fuzzy differential equations. Some numerical examp...

1 Introduction and basic definitions.

The construction of stiiy accurate and B-stable multi-implicit Runge-Kutta methods for parallel implementation is discussed. A fth and a seventh order method is constructed and a promising numerical comparison with the eecient Radau5 code of E. Hairer and G. Wanner is conducted.

The Diophantine equation x n−1 x−1 = yq has four known solutions in integers x, y, q and n with |x |, |y|, q > 1 and n > 2. Whilst we expect that there are, in fact, no more solutions, such a result is well beyond current technology. In this paper, we prove that if (x, y, n, q) is a solution to this equation, then n has three or fewer prime divisors, counted with multiplicity. This improves a r...