Related papers: Symmetry-preserving numerical schemes
We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE has notable invariant properties. An approximation scheme preserving the…
In this paper, we introduce second order and fourth order space discretization via finite difference implementation of the finite element method for solving Fokker-Planck equations associated with irreversible processes. The proposed…
A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. For this, a mathematical model is developed to incorporate homogeneous Dirichlet and Neumann type boundary conditions. The…
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,…
In this note we work on the construction of positive preserving numerical schemes for systems of stochastic differential equations. We use the semi discrete idea that we have proposed before proposing now a numerical scheme that preserves…
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 article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg-de…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
Ordinary differential equations (ODEs) and ordinary difference systems (O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$, $\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective…
We study numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature. We show that standard finite difference schemes are nonlinearly unstable. We build convergent finite…
We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.
This paper aims to develop a linearly implicit structure-preserving numerical scheme for the space fractional sine-Gordon equation, which is based on the newly developed invariant energy quadratization method. First, we reformulate the…
In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Lie…
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…
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…
A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to…
We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and self-gravitation modeling. The scheme is fully discrete and structure preserving in…
In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations…
Conservation laws are formulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not…
In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new…