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Motivated by the paraxial narrow-angle approximation of the Helmholtz equation in domains of variable topography that appears as an important application in Underwater Acoustics, we analyze a general Schr\"odinger-type equation posed on…
In this paper, we propose a numerical method for the solution of time-dependent flow problems in mixed form. Such problems can be efficiently approximated on hierarchical grids, obtained from an unstructured coarse triangulation by using a…
In the first days of quantum mechanics Dirac pointed out an analogy between the time-dependent coefficients of an expansion of the Schr\"odinger equation and the classical position and momentum variables solving Hamilton's equations. Here…
This work aims to construct an efficient and highly accurate numerical method to address the time singularity at $t=0$ involved in a class of time-fractional parabolic integro-partial differential equations in one and two dimensions. The…
This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schr\"{o}dinger equations with additional constraints. We include three…
The paper studies a time-nonlocal multiphysics finite element method with Crank-Nicolson scheme for poroelasticity model with secondary consolidation. For the case where the physical parameters $\lambda,\lambda^*$ and $c_0$ are all finite…
We study solutions to the Cauchy problem for the linear and nonlinear Schroedinger equation with a quadratic Hamiltonian depending on time. For the linear case the evolution operator can be expressed as an integral operator with the…
We present an exact three-dimensional wave solution to the shearing sheet equations of motion. The existence of this solution argues against transient amplification as a route to turbulence in unmagnetized disks. Moreover, because the…
This paper presents an analytical treatment of the path integral formalism for time-dependent quantum systems within the framework of Wigner-Dunkl mechanics, emphasizing systems with varying masses and time-dependent potentials. By…
A non-perturbative approach to the solution of the time-dependent, two-center Dirac equation is presented with a special emphasis on the proper treatment of the potential of the nuclei. In order to account for the full multipole expansion…
The validation and parallel implementation of a numerical method for the solution of the time-dependent Dirac equation is presented. This numerical method is based on a split operator scheme where the space-time dependence is computed in…
A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretisation and a stochastic trigonometric scheme for the temporal…
We investigate the approach of time-dependent variational principle (TDVP) for the one-dimensional spin-$J$ PXP model with detuning, which is relevant for programmable Rydberg atom arrays. The variational manifold is chosen as the minimally…
In this paper we consider a mass- and energy--conserving Crank-Nicolson time discretization for a general class of nonlinear Schr\"odinger equations. This scheme, which enjoys popularity in the physics community due to its conservation…
A new approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation is implemented. It is based on using Poisson's formula and its three time level modification combined with polynomial…
We consider time-dependent Schr\"{o}dinger equations for a free nonrelativistic particle on the three-dimensional $q$-deformed Euclidean space. We determine plane wave solutions to these Schr\"{o}dinger equations and show that they form a…
We obtain analytic solution of the time-independent Schrodinger equation in two dimensions for a charged particle moving in the field of an electric quadrupole. The solution is written as a series in terms of special functions that support…
Non-hydrostatic atmospheric models often use semi-implicit temporal discretisations in order to negate the time step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision…
In this paper, a fast solver is studied for saddle point system arising from a second-order Crank-Nicolson discretization of an initial-valued parabolic PDE constrained optimal control problem, which is indefinite and ill-conditioned.…
We propose and analyze a non-iterative domain decomposition integrator for the linear acoustic wave equation. The core idea is to combine an implicit Crank-Nicolson step on spatial subdomains with a local prediction step at the subdomain…