Related papers: Lie group stability of finite difference schemes
Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations…
Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and…
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations. This family of schemes has a simple form and can systemically achieve arbitrary high-order accuracy in theory. Moreover, these schemes can…
We present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail. Although numerical methods for geometric flows have been used frequently in axisymmetric settings,…
This paper describes a class of scheme named "residual distribution schemes" or "fluctuation splitting schemes". They are a generalization of Roe's numerical flux in fluctuation form. The so-called multidimensional fluctuation schemes have…
We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including…
In this paper, an energy-consistent finite difference scheme for the compressible hydrodynamic and magnetohydrodynamic (MHD) equations is introduced. For the compressible magnetohydrodynamics, an energy-consistent finite difference…
In the given paper we consider finite difference approximations to systems of polynomially-nonlinear partial differential equations whose coefficients are rational functions over rationals in the independent variables. The notion of strong…
We present a method to obtain symmetries for second-order systems of ordinary difference equations and how to use them to reduce the order. We also introduce a technique of finding conservation laws for such systems.
An energy stable finite element scheme within arbitrary Lagrangian Eulerian (ALE) framework is derived for simulating the dynamics of millimetric droplets in contact with solid surfaces. Supporting surfaces considered may exhibit…
It is shown that mathematical physics differential equations have properties that allow describing processes such as the structures emergence, discrete transitions, quantum jumps. The peculiarity is that such properties are hidden. They do…
A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and…
We construct and analyze a strongly consistent second-order finite difference scheme for the steady two-dimensional Stokes flow. The pressure Poisson equation is explicitly incorporated into the scheme. Our approach suggested by the first…
In this work we argue about the Lesche stability of some systems, that are motivated by the use of fractional derivatives.
Two-fluid relativistic plasma flow equations combine the equations of relativistic hydrodynamics with Maxwell's equations for electromagnetic fields, which involve divergence constraints for the magnetic and electric fields. When developing…
We study the construction of a non-standard finite differences numerical scheme for a general class of two dimensional differential equations including several models in population dynamics using the idea of non-local approximation…
Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems.…
In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic L\'evy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of…
We formulate a general criterion for the exact preservation of the "lake at rest" solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. The main idea is a…
In this work, we explore various relevant aspects of the Smoothed Particle Hydrodynamics regarding Burger's equation. The stability, precision, and efficiency of the algorithm are investigated in terms of different implementations. In…