Related papers: Computation of Maxwell's equations on manifold usi…
Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on…
High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on…
Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an…
The present paper is focused on the proof of the convergence of the discrete implicit Marker-and-Cell (MAC) scheme for time-dependent Navier--Stokes equations with variable density and variable viscosity. The problem is completed with…
We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations discretized on staggered (MAC) grids. The scheme is based on finite difference approximations…
This paper focuses on the question of how unconditional stability can be achieved via multistep ImEx schemes, in practice problems where both the implicit and explicit terms are allowed to be stiff. For a class of new ImEx multistep schemes…
A method to compute guaranteed lower bounds to the eigenvalues of the Maxwell system in two or three space dimensions is proposed as a generalization of the method of Liu and Oishi [SIAM J. Numer. Anal., 51, 2013] for the Laplace operator.…
An implicit multiscale method with multiple macroscopic prediction for steady state solutions of gas flow in all flow regimes is presented. The method is based on the finite volume discrete velocity method (DVM) framework. At the cell…
We address a numerical methodology for the computation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a "good" macroscopic description in the form of…
We present two strategies for designing passivity preserving higher order discretization methods for Maxwell's equations in nonlinear Kerr-type media. Both approaches are based on variational approximation schemes in space and time. This…
For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems…
In this article we have illustrated how is possible to formulate Maxwell's equations in vacuum in an independent form of the usual systems of units. Maxwell's equations, are then specialized to the most commonly used systems of units:…
We develop a numerical scheme for solving time-domain Maxwell's equation. The method is motivated by CIP method which uses function values and its derivatives as unknown variables. The proposed scheme is developed by using the Poisson…
We study the systematic numerical approximation of Maxwell's equations in dispersive media. Two discretization strategies are considered, one based on a traditional leapfrog time integration method and the other based on convolution…
Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a…
In exterior calculus on smooth manifolds, the exterior derivative and wedge product are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial…
Given a stochastic differential equation (SDE) in $\mathbb{R}^n$ whose solution is constrained to lie in some manifold $M \subset \mathbb{R}^n$, we propose a class of numerical schemes for the SDE whose iterates remain close to $M$ to high…
If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as…
The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well…
This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered…