Related papers: Conservative Integrators for Piecewise Smooth Syst…
We present a family of multistep integrators based on the Adams-Bashforth methods. These schemes can be constructed for arbitrary convergence order with arbitrary step size variation. The step size can differ between different subdomains of…
Conservative symmetric second-order one-step integrators are derived using the Discrete Multiplier Method for a family of vortex-blob models approximating the incompressible Euler's equations on the plane. Conservative properties and second…
We show how to increase the order of one-dimensional discrete gradient numerical integrator without losing its advantages, such as exceptional stability, exact conservation of the energy integral and exact preservation of the trajectories…
We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our…
In the article, we discuss the conservation laws for the nonlinear Schr\"{o}dinger equation with wave operator under multisymplectic integrator (MI). First, the conservation laws of the continuous equation are presented and one of them is…
Coarse-grained models that preserve hydrodynamics provide a natural approach to study collective properties of soft-matter systems. Here, we demonstrate that commonly used integration schemes in dissipative particle dynamics give rise to…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
Gaussian process regression is increasingly applied for learning unknown dynamical systems. In particular, the implicit quantification of the uncertainty of the learned model makes it a promising approach for safety-critical applications.…
Conservative symmetric second-order one-step schemes are derived for dynamical systems describing various many-body systems using the Discrete Multiplier Method. This includes conservative schemes for the $n$-species Lotka-Volterra system,…
In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schr\"{o}dinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence…
We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…
We present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. Given a system of differential equations possessing conservation laws, our approach is based on…
Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important…
In this paper we develop a conservative sharp-interface method dedicated to simulating multiple compressible fluids. Numerical treatments for a cut cell shared by more than two materials are proposed. First, we simplify the interface…
Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are…
In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…
We present a high-order method for flow simulation on unstructured curved nonconforming sliding meshes. This method utilizes dynamic transfinite mortar elements to exchange flow information between the two sides of a sliding interface. The…
In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry…
Impulse methods are generalized to a family of integrators for Langevin systems with quadratic stiff potentials and arbitrary soft potentials. Uniform error bounds (independent from stiff parameters) are obtained on integrated positions…