arXiv subjects

James Woodfield

Publications and source records attributed to James Woodfield.

9 recordsLinked to original sources

arXiv2026

Higher Order Multidimensional Slope Limiters with Local Maximum Principles

Higher-order numerical methods are used to find accurate numerical solutions to hyperbolic partial differential equations. Limiting is required to either converge to the correct type of solution or to adhere to physically motivated local maximum principles and less restrictive limiting procedures are required so as to not severely decrease the accuracy. In this paper, we adapt the existing slope limiter framework introduced in [Zhang \& Shu, J. Comput. Phys., 229(9):3091-3120, 2010] to achieve distinct local boundedness principles. We conclude that quadrature points contributing to numerical fluxes on either side of a face can be limited based on shared face-defined maximum principles and the resulting cell mean at the next timestep satisfies a cell mean maximum principle. Furthermore additional points arising in a decomposition of a cell mean must be limited locally when going beyond piecewise linear reconstructions. This allows the design of new multidimensional limiters which at second order can attain the same cell mean maximum principle as existing slope limiters, but allows more of the higher order flux to be used, generalises beyond second order schemes and can be modified for user specified local maximum principles.

math.NA↗

arXiv2025

Trajectory learning for ensemble forecasts via the continuous ranked probability score: a Lorenz '96 case study

This paper demonstrates the feasibility of trajectory learning for ensemble forecasts by employing the continuous ranked probability score (CRPS) as a loss function. Using the two-scale Lorenz '96 system as a case study, we develop and train both additive and multiplicative stochastic parametrizations to generate ensemble predictions. Results indicate that CRPS-based trajectory learning produces parametrizations that are both accurate and sharp. The resulting parametrizations are straightforward to calibrate and outperform derivative-fitting-based parametrizations in short-term forecasts. This approach is particularly promising for data assimilation applications due to its accuracy over short lead times.

math.NA↗

arXiv2025

Numerically modelling semidirect product geodesics

This paper numerically investigates Euler-Poincaré equations arising from a self-semidirect product group structure. Nonlinearly coupled systems of equations emerge from the semidirect product action where one set of dynamics can be considered in the frame of another. A monolithic energy-preserving continuous Galerkin finite element method is used to study geodesic equations associated with the semidirect product of the diffeomorphism group on a circle with itself. Theoretically predicted peakon solutions are observed as an emergent behaviour. In addition, complicated nonlinear transfers of energy are associated with the semidirect product coupling, where amongst various nonlinear interactions, we observe coupled peakon behaviour. A mimetic (C-grid) finite difference method is used to study the geodesic flow of the semidirect product of the volume preserving diffeomorphism group with itself, where similar coupling behaviour is observed in the vorticity variables. We also investigate coadjoint and Lie-Poisson structures in the context of geodesic equations on semidirect product groups, where the underlying group is first extended by central extension or semidirect product.

nlin.PS↗

arXiv2025

Lévy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics

Stochastic modelling necessitates an interpretation of noise. In this paper, we describe the loss of deterministically stable behaviour in a fundamental fluid mechanics problem, conditional to whether noise is introduced in the sense of Itô, Stratonovich or a limit of Wong-Zakai type. We examine this comparison in the wider context of discretising stochastic differential equations with and without the Lévy area. From the numerical viewpoint, we demonstrate performing higher order discretisations with the use of a Lévy area can lead to the loss of conserved area and angle quantities. Such behaviour is not physically expected in the Stratonovich model. Conversely, we study Stochastic Advection by Lie Transport and its derivation from homogenisation theory, which introduces drift corrections of the same class naturally. From the viewpoint of homogenisation, the qualitative properties of the Wong-Zakai anomaly are physically motivated as arising due to correlations from a fast and mean scale fluid decomposition.

math.DS↗

arXiv2025

Monotone conservative strategies in data assimilation

This paper studies whether numerically preserving monotonic properties can offer modelling advantages in data assimilation, particularly when the signal or data is a realization of a stochastic partial differential equation (SPDE) or partial differential equation (PDE) with a monotonic property. We investigate the combination of stochastic Strong Stability Preserving (SSP) time-stepping, nonlinear solving strategies and data assimilation. Experimental results indicate that a particle filter whose ensemble members are solved monotonically can increase forecast skill when the reference data (not necessarily observations) also has a monotone property. Additionally, more advanced techniques used to avoid the degeneracy of the filter (tempering-jittering) are shown to be compatible with a conservative monotone approach.

physics.comp-ph↗

arXiv2024

Strong Stability Preservation for Stochastic Partial Differential Equations

This paper extends deterministic notions of Strong Stability Preservation (SSP) to the stochastic setting, enabling nonlinearly stable numerical solutions to stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) with pathwise solutions that remain unconditionally bounded. This approach may offer modelling advantages in data assimilation, particularly when the signal or data is a realization of an SPDE or PDE with a monotonicity property.

math.NA↗

arXiv2024

Comparing two different types of stochastic parametrisation in geophysical flow

This paper investigates the effects of stochastic variations in bathymetry on the solutions of the thermal quasigeostrophic (TQG) equations. These stochastic perturbations generate a variety of different types of ensemble spread in the solution behaviour whilst also preserving the deterministic Lie Poisson structure and Casimir conservation laws. We numerically compare the solution sensitivity, to another type of structure-preserving stochastic perturbation where instead of bathymetry, the velocity is stochastically perturbed.

physics.flu-dyn↗

arXiv2024

New limiter regions for multidimensional flows

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms. The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection.

math.NA↗

arXiv2024

Stochastic fluids with transport noise: Approximating diffusion from data using SVD and ensemble forecast back-propagation

We introduce and test methods for the calibration of the diffusion term in Stochastic Partial Differential Equations (SPDEs) describing fluids. We take two approaches, one uses ideas from the singular value decomposition and the Biot-Savart law. The other backpropagates through an ensemble forecast, with respect to diffusion parameters, to minimise a probabilistic ensemble forecasting metric. We describe the approaches in the specific context of solutions to SPDEs describing the evolution of fluid particles, sometimes called inviscid vortex methods. The methods are tested in an idealised setting in which the reference data is a known realisation of the parameterised SPDE, and also using a forecast verification metric known as the Continuous Rank Probability Score (CRPS).

physics.flu-dyn↗