Related papers: Lie group stability of finite difference schemes
A method for enhancing the stability and robustness of explicit schemes in computational fluid dynamics is presented. The method is based in reformulating explicit schemes in matrix form, which cane modified gradually into semi or…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…
We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish $L^2$ stability and convergence of the difference approximations under conditions that are less strict than those required…
The finite-difference time-domain (FDTD) algorithm is a popular numerical method for solving electromagnetic problems. FDTD simulations can suffer from instability due to the explicit nature of the method. Stability enforcement can be…
Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and…
In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of…
The aim of this work is to develop general optimization methods for finite difference schemes used to approximate linear differential equations. The specific case of the transport equation is exposed. In particular, the minimization of the…
Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
The monotonicity and stability of difference schemes for, in general, hyperbolic systems of conservation laws with source terms are studied. The basic approach is to investigate the stability and monotonicity of a non-linear scheme in terms…
A class of finite difference schemes for solving a fractional anti-diffusive equation, recently proposed by Andrew C. Fowler to describe the dynamics of dunes, is considered. Their linear stability is analyzed using the standard Von Neumann…
We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case in certain discrete…
This paper provides new summation inequalities in both single and double forms to be used in stability analysis of discrete-time systems with time-varying delays. The potential capability of the newly derived inequalities is demonstrated by…
We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces,…
In this study I develop a novel action for lattice gauge theory for finite systems, which accommodates non-periodic boundary conditions, implements the proper integral form of Gauss' law and exhibits an inherently symmetric energy momentum…
The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics,…
The conservation laws of continuum mechanic written in an Eulerian frame make no difference between fluids and solids except in the expression of the stress tensors, usually with Newton's hypothesis for the fluids and Helmholtz potentials…
In this work we consider an extension of a recently proposed structure preserving numerical scheme for nonlinear Fokker-Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are…