Related papers: Energy dissipative numerical scheme for gradient f…
In this paper, we consider numerical approximation of constrained gradient flows of planar closed curves, including the Willmore and the Helfrich flows. These equations have energy dissipation and the latter has conservation properties due…
We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting…
The equality between dissipation and energy drop is a structural property of gradient-flow dynamics. The classical implicit Euler scheme fails to reproduce this equality at the discrete level. We discuss two modifications of the Euler…
We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial…
The notion of dissipative dynamical systems provides a formal description of processes that cannot generate energy internally. For these systems, changes in energy can only occur due to an external energy supply or dissipation effects.…
We present a systematical approach to developing arbitrarily high order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy…
In this paper, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and…
In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large…
We propose a new semi-discretization scheme to approximate nonlinear Fokker-Planck equations, by exploiting the gradient flow structures with respect to the 2-Wasserstein metric. We discretize the underlying state by a finite graph and…
We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$,…
We address surface gradient flows which allow for energy dissipation by evolving the surface and a scalar quantity on it, simultaneously. A proper choice of the time derivative and the gauge of surface independence guarantees energy…
We present a new temporal discretization paradigm for developing energy-production-rate preserving numerical approximations to thermodynamically consistent partial differential equation systems, called the supplementary variable method. The…
Numerical simulation of compressible fluid flows is performed using the Euler equations. They include the scalar advection equation for the density, the vector advection equation for the velocity and a given pressure dependence on the…
Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient…
This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for…
In this paper, we conduct an in-depth investigation of the structural intricacies inherent to the Invariant Energy Quadratization (IEQ) method as applied to gradient flows, and we dissect the mechanisms that enable this method to uphold…
We propose and analyze stable finite element approximations for Willmore flow of planar curves. The presented schemes are based on a novel weak formulation which combines an evolution equation for curvature with the curvature formulation…
We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the general gradient flow…
We present an effective numerical procedure, which is based on the computational scheme from [Heid et al., arXiv:1906.06954], for the numerical approximation of excited states of Schr\"odingers equation. In particular, this procedure…
In this paper we present a complete framework for the energy-stable simulation of stratified incompressible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid…