Numerical integrators for motion under a strong constraining force

This paper deals with the numerical integration of Hamiltonian systems in which a stiff anharmonic potential causes highly oscillatory solution behavior with solution-dependent frequencies. The impulse method, which uses micro- and macro-steps for the integration of fast and slow parts, respectively, does not work satisfactorily on such problems. Here it is shown that variants of the impulse method with suitable projection preserve the actions as adiabatic invariants and yield accurate approximations, with macro-stepsizes that are not restricted by the stiffness parameter

Space-time adaptive resolution for reactive flows

Multi-scale systems evolve over a wide range of temporal and spatial scales. The extent of time scales makes both theoretical and numerical analysis difficult, mostly because the time scales of interest are typically much slower than the fastest scales occurring in the system. Systems with such characteristics are usually classified as being stiff. An adaptive mesh refinement method based on the wavelet transform and the G-Scheme framework are used to achieve spatial and temporal adaptive model reduction, respectively, of physical problems described by PDEs. The combination of the methods is proposed to solve PDEs describing reaction-diffusion systems with the minimal number of degrees of freedom, for prescribed accuracies in space and time. Different reaction-diffusion systems are studied with the aim to test the performance and the capability of the combined scheme to generate accurate solutions with respect to reference ones. Several strategies are implemented to improve the performance of the scheme, with minimal loss of accuracy

Variational Bonded Discrete Element Method with Manifold Optimization

This paper proposes a novel approach that combines variational integration with the bonded discrete element method (BDEM) to achieve faster and more accurate fracture simulations. The approach leverages the efficiency of implicit integration and the accuracy of BDEM in modeling fracture phenomena. We introduce a variational integrator and a manifold optimization approach utilizing a nullspace operator to speed up the solving of quaternion-constrained systems. Additionally, the paper presents an element packing and surface reconstruction method specifically designed for bonded discrete element methods. Results from the experiments prove that the proposed method offers 2.8 to 12 times faster state-of-the-art methods

Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods