Related papers: Difference schemes with point symmetries and their…
In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for…
An important problem that arises in many engineering applications is the boundary value problem for ordinary differential equations. There have been many computational methods proposed for dealing with this problem. The convergence of the…
There is a qualitative difference between one-dimensional and multi-dimensional solutions to the Euler equations: new features that arise are vorticity and a nontrivial incompressible (low Mach number) limit. They present challenges to…
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…
In the deterministic realm, both differential equations and symmetry generators are geometrical objects, and behave properly under changes of coordinates; actually this property is essential to make symmetry analysis independent of the…
In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform…
In ordinary turbulence research it has been a long standing tradition to solve the equations in spectral space giving the best possible accuracy. This is indeed a natural choice for incompressible problems with periodic boundaries, but it…
In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modeling of poroelasticity processes. Stability estimates of weighted difference schemes for the coupled system of equations…
Two-fluid plasma flow equations describe the flow of ions and electrons with different densities, velocities, and pressures. We consider the ideal plasma flow i.e. we ignore viscous, resistive, and collision effects. The resulting system of…
We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by…
Nonclassical symmetries and reductions of polynomial equations and systems of polynomial equations are considered. It is shown that specific polynomial equations having "hidden" symmetries can be reduced to classical symmetric systems of…
We show a novel systematic way to construct conservative finite difference schemes for quasilinear first-order system of ordinary differential equations with conserved quantities. In particular, this includes both autonomous and…
The Inverse Problem for the estimation of a point-wise approximation error occurring at the discretization and solving of the system of partial differential equations is addressed. The set of the differences between the numerical solutions…
Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to…
Symmetry is an important factor in solving many constraint satisfaction problems. One common type of symmetry is when we have symmetric values. In a recent series of papers, we have studied methods to break value symmetries. Our results…
Preserving scalar boundedness is important for numerical schemes used in turbulent compressible multi-component flow simulations to prevent unphysical results and unstable simulations. However, ensuring scalar boundedness for high-order,…
This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the…
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…
We compare some first order well-balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography. We show that the space step required to obtain a prescribed…
Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…