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We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete…
A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion…
We propose an exponential integrator for the drift-kinetic equation in cylindrical geometry. This approach removes the CFL condition from the linear part of the system (which is often the most stringent requirement in practice) and treats…
We present a systematical approach to developing arbitrarily high order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy…
We consider a class of finite element approximations for fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. In our approach, we first solve a variational problem…
A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new…
In this paper, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and…
This paper introduces an efficient high-order numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Building upon the ideas from [Arnold, Ben Abdallah, Negulescu, SIAM J. Numer. Anal.,…
In this paper we extend the polynomial time integration framework to include exponential integration for both partitioned and unpartitioned initial value problems. We then demonstrate the utility of the exponential polynomial framework by…
In this paper, we prove that a class of autonomous piecewise continuous systems of fractional order has well-defined Lyapunov exponents. For this purpose, based on some known results from differential inclusions of integer and fractional…
We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics,…
Semi-implicit spectral deferred correction (SDC) methods provide a systematic approach to construct time integration methods of arbitrarily high order for nonlinear evolution equations including conservation laws. They converge towards $A$-…
As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(\tau^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on…
We consider initial-boundary-value problems for a class of nonlinear third order equations having non-autonomous forcing terms and get new asymptotic stability results by means of the Liapunov second method. The class includes equations…
A recently developed method for the calculation of Lyapunov exponents of dynamical systems is described. The method is applicable whenever the linearized dynamics is Hamiltonian. By utilizing the exponential representation of symplectic…
We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our…
This paper is concerned with relationships of Lyapunov exponents with sensitivity and stability for non-autonomous discrete systems. Some new concepts are introduced for non-autonomous discrete systems, including Lyapunov exponents, strong…
The integrating factor and exponential time differencing methods are implemented and tested for solving the time-dependent Kohn--Sham equations. Popular time propagation methods used in physics, as well as other robust numerical approaches,…
This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the…
In this paper, we introduce second order and fourth order space discretization via finite difference implementation of the finite element method for solving Fokker-Planck equations associated with irreversible processes. The proposed…