Related papers: Forward Symplectic Integrators and the Long Time P…
We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove stability and derive 4th…
The simulation of systems that act on multiple time scales is challenging. A stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part, a coarser approximation is accurate…
In this paper, we present the two-step trigonometrically fitted symmetric Obrechkoff methods with algebraic order of twelve. The method is based on the symmetric two-step Obrechkoff method, with 12 algebraic order, high phase-lag order and…
We develop a fourth order simulation algorithm for solving the stochastic Langevin equation. The method consists of identifying solvable operators in the Fokker-Planck equation, factorizing the evolution operator for small time steps to…
We give an iterative algorithm for phase estimation of a parameter theta, which is within a logarithmic factor of the Heisenberg limit. Unlike other methods, we do not need any entanglement or an extra rotation gate which can perform…
Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the…
Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the…
We consider the dynamics of a parabolic and a hyperbolic equation coupled on a common interface and develop time-stepping schemes that can use different time-step sizes for each of the subproblems. The problem is formulated in a strongly…
This paper is devoted to the construction of exponential integrators of first and second order for the time discretization of constrained parabolic systems. For this extend, we combine well-known exponential integrators for unconstrained…
The Stoermer-Verlet-leapfrog group of integrators commonly used in molecular dynamics simulations has long become a textbook subject and seems to have been studied exhaustively. There are, however, a few striking effects in performance of…
Seismic imaging is a major challenge in geophysics with broad applications. It involves solving wave propagation equations with absorbing boundary conditions (ABC) multiple times. This drives the need for accurate and efficient numerical…
The solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data. Accurate solutions can be systematically derived by decomposing the exponential in a product form. For…
We derive a computable a posteriori error estimator for the $\alpha$-harmonic extension problem, which localizes the fractional powers of elliptic operators supplemented with Dirichlet boundary conditions. Our a posteriori error estimator…
We introduce a new family of symplectic integrators depending on a real parameter. When the paramer is zero, the corresponding method in the family becomes the classical Gauss collocation formula of order 2s, where s denotes the number of…
In this paper, we consider the numerical approximation of time-fractional parabolic problems involving Caputo derivatives in time of order $\alpha$, $0< \alpha<1$. We derive optimal error estimates for semidiscrete Galerkin FE type…
We present a new mixed variable symplectic (MVS) integrator for planetary systems, that fully resolve close encounters. The method is based on a time regularisation that allows keeping the stability properties of the symplectic integrators,…
This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental…
For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well…
In this note, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in [Sakamoto-van~der Schaft, IEEE Transactions on Automatic Control, 2008]. Our algorithm…
The symplectic Wisdom-Holman map revolutionized long-term integrations of planetary systems. There is freedom in such methods of how to split the Hamiltonian and which coordinate system to employ, and several options have been proposed in…