Related papers: A $\mu$-mode integrator for solving evolution equa…
This article deals with the numerical integration in time of nonlinear Schr\"odinger equations. The main application is the numerical simulation of rotating Bose-Einstein condensates. The authors perform a change of unknown so that the…
Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems,…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…
Exponential integrators are explicit methods for solving ordinary differential equations that treat linear behaviour exactly. The stiff-order conditions for exponential integrators derived in a Banach space framework by Hochbruck and…
This paper provides an algebraic framework for the generation of order conditions for the construction of exponential integrators like splitting and Magnus-type methods for the numerical solution of evolution equations. The generation of…
We describe an interface and an implementation for performing Kronecker product actions on NVIDIA GPUs for multiple small 2-D matrices and 3-D arrays processed in parallel as a batch. This method is suited to cases where the Kronecker…
A new Active Flux method for the multi-dimensional Euler equations is based on an additive operator splitting into acoustics and advection. The acoustic operator is solved in a locally linearized manner by using the exact evolution…
Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute…
A quantitative definition of numerical stiffness for initial value problems is proposed. Exponential integrators can effectively integrate linearly stiff systems, but they become expensive when the linear coefficient is a matrix, especially…
Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this…
Recently a splitting approach has been presented for the simulation of sonic-boom propagation. Splitting methods allow one to divide complicated partial differential equations into simpler parts that are solved by specifically tailored…
The simulation of multi-body systems with frictional contacts is a fundamental tool for many fields, such as robotics, computer graphics, and mechanics. Hard frictional contacts are particularly troublesome to simulate because they make the…
We construct numerical integrators for Hamiltonian problems that may advantageously replace the standard Verlet time-stepper within Hybrid Monte Carlo and related simulations. Past attempts have often aimed at boosting the order of accuracy…
We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction,…
We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our…
Most numerical methods for time integration use real-valued time steps. Complex time steps, however, can provide an additional degree of freedom, as we can select the magnitude of the time step in both the real and imaginary directions. We…
The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the…
Convolution-type integral equations arise from various fields, \textit{e.g.}, finite impulse response filters in signal processing and deblurring problems in image processing. When solving these equations, conventional numerical methods,…
We present a novel hierarchical formulation of the fourth-order forward symplectic integrator and its numerical implementation in the GPU-accelerated direct-summation N-body code FROST. The new integrator is especially suitable for…
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…