Related papers: Symmetry-preserving numerical schemes
A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified…
The one-dimensional shallow water equations in Eulerian and Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian (potential) coordinates and symmetries and conservation…
Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or…
This article is concerned with the derivation of numerical reconstruction schemes for the inverse moving source problem on determining source profiles in (time-fractional) evolution equations. As a continuation of the theoretical result on…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
In this paper we present a simple and accurate second order finite element scheme to simulate the Burgers' equation on the whole real line and subjected to initial conditions with compact support. The numerical simulations are performed by…
We propose a novel formulation for parametric finite element methods to simulate surface diffusion of closed curves, which is also called as the curve diffusion. Several high-order temporal discretizations are proposed based on this new…
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…
For a field theory that is invariant under diffeomorphisms there is a subtle interplay between symmetries, conservation laws and the phase space of the theory. The natural language for describing these ideas is that of differential forms…
We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of…
Symmetry principles underlie and guide scientific theory and research, from Curie's invariance formulation to modern applications across physics, chemistry, and mathematics. Building on a recent matrix Lie group measurement model, this…
We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on…
Entropy conservation and stability of numerical methods in gas dynamics have received much interest. Entropy conservative numerical fluxes can be used as ingredients in two kinds of schemes: Firstly, as building blocks in the subcell flux…
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 article we describe applications of Discrete Differential Forms in computational GR. In particular we consider the initial value problem in vacuum space-times that are spherically symmetric. The motivation to investigate this method…
We develop the novel method of artificial barriers for scalar stochastic differential equations (SDEs) and use it to construct boundary-preserving numerical schemes for strong approximation of scalar SDEs, possibly with non-globally…
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…
Kahan's method and a two-step generalization of the discrete gradient method are both linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a cubic Hamiltonian. These methods are here investigated and…
We consider the calculation of Euler--Lagrange systems of ordinary difference equations, including the difference Noether's Theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We…
This paper investigates the structure preservation and convergence analysis of a class of fully discrete finite difference schemes for the stochastic heat equation driven by L\'evy space-time white noise. The novelty lies in the…