Related papers: Discrete gradient methods have an energy conservat…
The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of…
Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space-time scale. In these…
Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow.…
In this work we present a nonlocal conservation law with a velocity depending on an integral term over a part of the space. The model class covers already existing models in literature, but it is also able to describe new dynamics mainly…
Conservation laws are of great theoretical and practical interest. We describe a novel approach to machine learning conservation laws of finite-dimensional dynamical systems using trajectory data. It is the first such approach based on…
We present linearly implicit methods that preserve discrete approximations to local and global energy conservation laws for multi-symplectic PDEs with cubic invariants. The methods are tested on the one-dimensional Korteweg-de Vries…
Structure-preserving algorithms for solving conservative PDEs with added linear dissipation are generalized to systems with time-dependent damping/driving terms. This study is motivated by several PDE models of physical phenomena, such as…
We introduce an energy-based model, which seems especially suited for constrained systems. The proposed model provides an alternative to the popular port-Hamiltonian framework and exhibits similar properties such as energy dissipation as…
Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux…
The fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high- order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain…
An energy-based discontinuous Galerkin method for the advective wave equation is proposed and analyzed. Energy-conserving or energy-dissipating methods follow from simple, mesh-independent choices of the inter-element fluxes, and both…
We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built…
This paper proposes four fundamental requirements for establishing PDEs (partial differential equations) modeling irreversible processes. We show that the PDEs derived via the CDF (conservation-dissipation formalism) meet all the…
This paper introduces a family of entropy-conserving finite-difference discretizations for the compressible flow equations. In addition to conserving the primary quantities of mass, momentum, and total energy, the methods also preserve…
In this paper, a novel dual-field structure-preserving mixed finite element discretization for incompressible Hall MHD equations is introduced. The discretization satisfies pointwise conservation of mass, magnetic Gauss's law, and…
The spatial discretization of convective terms in compressible flow equations is studied from an abstract viewpoint, for finite-difference methods and finite-volume type formulations with cell-centered numerical fluxes. General conditions…
Conservation of current and conservation of charge are nearly the same thing: when enough is known about charge movement, conservation of current can be derived from conservation of charge, in ideal dielectrics, for example. Conservation of…
All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted…
The discrete gradient approach is generalized to yield integral preserving methods for differential equations in Lie groups.
Time-space noncommutativity leads to quantisation of time and energy nonconservation when time is conjugate to a compact spatial direction like a circle. In this context energy is conserved only modulo some fixed unit. Such a possibility…