Related papers: Revisiting high-order Taylor methods for astrodyna…
We design a novel, exactly energy-conserving implicit non-symplectic integration method for an eight-dimensional Hamiltonian system with four degrees of freedom. In our algorithm, each partial derivative of the Hamiltonian with respect to…
In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis…
We develop new conservative discontinuous Galerkin (DG) methods for nonlinear wave problems, focusing on the generalized Korteweg-de Vries (gKdV) equation and the coupled Hirota-Satsuma KdV (HS-KdV) system. The proposed methods preserve…
We propose a new family of high-order explicit generalized-$\alpha$ methods for hyperbolic problems with the feature of dissipation control. Our approach delivers $2k,\, \left(k \in \mathbb{N}\right)$ accuracy order in time by solving $k$…
We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these methods as well as all data files…
Adaptive optics images from the W. M. Keck Observatory have delivered numerous influential scientific results, including detection of multi-system asteroids, the supermassive black hole at the center of the Milky Way, and directly imaged…
A new method is given for the model-space effective interaction. Introducing a new operator in place of the Q-box in the Krenciglowa-Kuo (KK) method, we derive a new equation for the effective interaction. This equation can be viewed as an…
We carry out a stability and convergence analysis of a fully discrete scheme for the time-dependent Navier-Stokes equations resulting from combining an $H(\mathrm{div}, \Omega)$-conforming discontinuous Galerkin spatial discretization, and…
We consider the numerical discretization of the time-domain Maxwell's equations with an energy-conserving discontinuous Galerkin finite element formulation. This particular formulation allows for higher order approximations of the electric…
This chapter provides an introduction to Hybrid High-Order (HHO) methods. These are new generation numerical methods for PDEs with several advantageous features: the support of arbitrary approximation orders on general polyhedral meshes,…
We discuss the dynamics and thermodynamics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system to the canonical description of a stochastically forced Brownian system. We…
We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity…
Recently a new class of numerical integration methods -- ``mixed variable symplectic integrators'' -- has been introduced for studying long-term evolution in the conservative gravitational few-body problem. These integrators are an order of…
This is a short review of a series of papers which, in collaboration with Yue Ma, establish several novel existence results for systems of coupled wave-Klein-Gordon equation. Our method, the Hyperbolic Hyperboloidal Method, has allowed us…
In this paper we develop a higher-order method for solving composite (non)convex minimization problems with smooth (non)convex functional constraints. At each iteration our method approximates the smooth part of the objective function and…
Well tempering is one of the few classical field theory methods for solving the original cosmological constant problem, dynamically canceling a large (possibly Planck scale) vacuum energy and leaving the matter component intact, while…
We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets,…
To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard-Lindel\"of iteration), where at each iteration a linear inhomogeneous…
While cosmological dark matter-only simulations relying solely on gravitational effects are comparably fast to compute, baryonic properties in simulated galaxies require complex hydrodynamic simulations that are computationally costly to…
Numerical evolution of time-dependent differential equations via explicit Runge-Kutta or Taylor methods typically fails to preserve symmetries of a system. It is known that there exists no numerical integration method that in general…