Related papers: Approximate Solutions to Second-Order Parabolic Eq…
We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $\Omega \subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that…
Many applications using large datasets require efficient methods for minimizing a proximable convex function subject to satisfying a set of linear constraints within a specified tolerance. For this task, we present a proximal projection…
This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave…
Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double time Green's function into Taylor series of the parameter $\lambda$ in the Hamiltonian $H=H_0 + \lambda H_1$. Here $H_0$ is the…
We discuss differential-- versus integral--equation based methods describing out--of thermal equilibrium systems and emphasize the importance of a well defined reduction to statistical observables. Applying the projection operator approach,…
The implicit compact finite-difference scheme was developed for evolutionary partial differential parabolic and Schr\"odinger-type equations and systems with a weak nonlinearity. To make a temporal step of the compact implicit scheme we…
This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial conditions. The class of…
The nonequilibrium Green's function (NEGF) formalism is a powerful tool to study the nonequilibrium dynamics of correlated lattice systems, but its applicability to realistic system sizes and long timescales is limited by unfavorable memory…
We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear…
This work is devoted to the study of a posteriori error estimation and adaptivity in parabolic problems with a particular focus on spatial discontinuous Galerkin (dG) discretisations. We begin by deriving an a posteriori error estimator for…
This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped…
This report presents a low computational and cognitive complexity, stable, time accurate and adaptive method for the Navier-Stokes equations. The improved method requires a minimally intrusive modification to an existing program based on…
Real data are constrained to finite sampling rates, which calls for a suitable mathematical description of the corrections to the finite-time estimations of the dynamic equations. Often in the literature, lower order discrete time…
In a recent work, we presented the first application of the Poisson-Wiseman-Anderson method of `matched expansions' to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the…
We propose a general strategy to discretize the Dyson series without applying direct numerical quadrature to high-dimensional integrals, and extend this framework to open quantum systems. The resulting discretization can also be interpreted…
One of the outstanding problems in the numerical discretization of the Feynman-Kac formula calls for the design of arbitrary-order short-time approximations that are constructed in a stable way, yet only require knowledge of the potential…
This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady…
Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution…
The two-particle problem within a nonequilibrium many-particle system is investigated in the framework of real-time Green's functions. Starting from the dynamically screened ladder approximation of the nonequlibrium Bethe-Salpeter equation,…
In this paper, approximate analytical solutions of nonlinear Emden-Fowler type equations are obtained by the differential transform method (DTM). The DTM is a numerical as well as analytical method for solving integral equations, ordinary…