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We develop and analyze a class of maximum bound preserving schemes for approximately solving Allen--Cahn equations. We apply a $k$th-order single-step scheme in time (where the nonlinear term is linearized by multi-step extrapolation), and…
We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and…
The objective of this work is the introduction and investigation of favourable time integration methods for the Gross--Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes…
A sequential quadratic programming method is designed for solving general smooth nonlinear stochastic optimization problems subject to expectation equality constraints. We consider the setting where the objective and constraint function…
In this paper, we present a rigorous proof of the convergence of first order and second order exponential time differencing (ETD) schemes for solving the nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the Fourier…
We study an explicit exponential scheme for the time discretisation of stochastic Schr\"odinger equations driven by additive or multiplicative Ito noise. The numerical scheme is shown to converge with strong order $1$ if the noise is…
Immersed boundary methods have attracted substantial interest in the last decades due to their potential for computations involving complex geometries. Often these cannot be efficiently discretized using boundary-fitted finite elements.…
We consider quadrature formulas of high order in time based on Radau-type, L-stable implicit Runge-Kutta schemes to solve time dependent stiff PDEs. Instead of solving a large nonlinear system of equations, we develop a method that performs…
In this paper we study a numerical implementation for the initial boundary value formulation for the generalized conformal field equations. We propose a formulation which is well suited for the study of the long-time behaviour of perturbed…
We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show…
This paper studies an infinite horizon optimal control problem for discrete-time linear system and quadratic criteria, both with random parameters which are independent and identically distributed with respect to time. In this general…
This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power…
This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has…
In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than…
In this paper, we propose a numerical method to approximate the solution of the time-dependent Schr\"odinger equation with periodic boundary condition in a high-dimensional setting. We discretize space by using the Fourier pseudo-spectral…
This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder.…
A local approach to the time integration of PDEs by exponential methods is proposed, motivated by theoretical estimates by A.Iserles on the decay of off-diagonal terms in the exponentials of sparse matrices. An overlapping domain…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…