Related papers: Parallelisation of partial differential equations …
In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the…
This article reviews many manifestations and applications of dual representations of pairs of groups, primarily in atomic and nuclear physics. Examples are given to show how such paired representations are powerful aids in understanding the…
The connection between symmetries and linearizations of discrete-time dynamical systems is being inverstigated. It is shown, that existence of semigroup structures related to the vector field and having linear representations enables…
Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact…
This paper is devoted to the study of reflected Stochastic Differential Equations when the constraint is not on the paths of the solution but acts on the law of the solution. These reflected equations have been introduced recently by…
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretisation of such problems does not necessarily yield Hamiltonian dynamics and…
Mathematical descriptions of flow phenomena usually come in the form of partial differential equations. The differential operators used in these equations may have properties such as symmetry, skew-symmetry, positive or negative…
Due to the Heisenberg uncertainty principle, various classical systems differing only on the scale smaller than Planck's cell correspond to the same quantum system. This fact is used to find a unique semiclassical representation without the…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
The paper deals with the developing of the methodological backgrounds for the modeling and simulation of complex dynamical objects. Such backgrounds allow us to perform coordinate transformation and formulate the algorithm of its usage for…
Given a differential equation with infinite-dimensional symmetry pseudo-group it is shown, using an example, that it is generally not possible to construct enough joint invariants to form an invariant numerical scheme of the equation. To…
A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…
In this article, we present a parallel recursive algorithm based on multi-level domain decomposition that can be used as a precondtioner to a Krylov subspace method to solve sparse linear systems of equations arising from the discretization…
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…
For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the…
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained…
Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…