Related papers: Structure-preserving integrators for the Benjamin-…
This work proposes a suite of numerical techniques to facilitate the design of structure-preserving integrators for nonlinear dynamics. The celebrated LaBudde-Greenspan integrator and various energy-momentum schemes adopt a difference…
Structure-preserving integrators are in the focus of ongoing research because of their distinguished features of robustness and long time stability. In particular, their formulation for coupled problems that include dissipative mechanisms…
We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible-irreversible coupling). We…
In this paper, we analyze finite difference schemes for Benjamin-Ono equation, u_t = uu_x + Hu_{xx}, where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data…
This paper proposes a new class of mass or energy conservative numerical schemes for the generalized Benjamin-Ono (BO) equation on the whole real line with arbitrarily high-order accuracy in time. The spatial discretization is achieved by…
We propose a linearly implicit structure-preserving numerical method for semilinear Hamiltonian systems with polynomial nonlinearities, combining Kahan's method and exponential integrator. This approach efficiently balances computational…
An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. Numerical integrators using generating functions, Hamiltonian splitting,…
We present novel geometric numerical integrators for Hunter--Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter--Saxton equation, the modified…
We present a new class of exponential integrators for ordinary differential equations. They are locally exact, i.e., they preserve the linearization of the original system at every point. Their construction consists in modifying existing…
We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on…
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…
We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent…
Methods for discretizing port-Hamiltonian systems are of interest both for simulation and control purposes. Despite the large literature on mixed finite elements, no rigorous analysis of the connections between mixed elements and…
The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe-Salpeter equation in the context of computing exciton energies and states. A computational challenge is that at least half of the…
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the…
We propose and analyze a structure-preserving approximation of the non-isothermal Cahn-Hilliard equation using conforming finite elements for the spatial discretization and a problem-specific mixed explicit-implicit approach for the…
We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and self-gravitation modeling. The scheme is fully discrete and structure preserving in…
Multi-symplectic integrators are typically regarded as a discretization of the Hamiltonian partial differential equations. This is due to the fact that, for generic finite-dimensional Hamiltonian systems, there exists only one independent…
The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and…
The H(div)-conforming approach for the Brinkman equation is studied numerically, verifying the theoretical a priori and a posteriori analysis in previous work of the authors. Furthermore, the results are extended to cover a non-constant…