Related papers: Symmetric-adjoint and symplectic-adjoint methods a…
We give an overview of the two different methods that have been introduced in order to describe the dynamics of constrained quantum systems; the symplectic formulation and the metric formulation. The symplectic method extends the work of…
Based on reasonable testing model problems, we study the preservation by symplectic Runge-Kutta method (SRK) and symplectic partitioned Runge-Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The…
A new geometric procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the…
Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics, with the isospectral property often corresponding to mathematically or physically…
We analyze composition methods with complex coefficients exhibiting the so-called ``symmetry-conjugate'' pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when…
Explicit integrating factor Runge-Kutta methods are attractive and popular in developing high-order maximum bound principle preserving time-stepping schemes for Allen-Cahn type gradient flows. However, they always suffer from the…
In this paper, exponential Runge-Kutta methods of collocation type (ERKC) which were originally proposed in (Appl Numer Math 53:323-339, 2005) are extended to semilinear parabolic problems with time-dependent delay. Two classes of the ERKC…
We have shown previously that functionally fitted Runge-Kutta (FRK) methods can be studied using a convenient collocation framework. Here, we extend that framework to functionally fitted Runge-Kutta-Nystr\"om (FRKN) methods, shedding…
A novel symplectic integrator for Hamiltonian equations on $S_2^n \times T^{\ast} \RR^m$ is developed and studied. Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltionian manifolds are studied, specifically,…
Mixed precision Runge--Kutta methods have been recently developed and used for the time-evolution of partial differential equations. Two-derivative Runge--Kutta schemes may offer enhanced stability and accuracy properties compared to…
Many control, optimization, and learning algorithms rely on discretizations of continuous-time contracting systems, where preservation of contractivity under numerical integration is key for stability, robustness, and reliable fixed-point…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be…
We introduce a class of general purpose linear multisymplectic integrators for Hamiltonian wave equations based on a diamond-shaped mesh. On each diamond, the PDE is discretized by a symplectic Runge--Kutta method. The scheme advances in…
The generalized additive Runge-Kutta (GARK) framework provides a powerful approach for solving additively partitioned ordinary differential equations. This work combines the ideas of symplectic GARK schemes and multirate GARK schemes to…
Different families of Runge-Kutta-Nystr\"om (RKN) symplectic splitting methods of order 8 are presented for second-order systems of ordinary differential equations and are tested on numerical examples. They show a better efficiency than…
Hamilton's equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose,…
The new class of alternating-conjugate splitting methods is presented and analyzed. They are obtained by concatenating a given composition involving complex coefficients with the same composition but with the complex conjugate coefficients.…
This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low-rank…