C. C. Pain

T. R. F. Phillips, C. E. Heaney, C. Boyang et al.

This paper presents a new approach which uses the tools within Artificial Intelligence (AI) software libraries as an alternative way of solving partial differential equations (PDEs) that have been discretised using standard numerical methods. In particular, we describe how to represent numerical discretisations arising from the finite volume and finite element methods by pre-determining the weights of convolutional layers within a neural network. As the weights are defined by the discretisation scheme, no training of the network is required and the solutions obtained are identical (accounting for solver tolerances) to those obtained with standard codes often written in Fortran or C++. We also explain how to implement the Jacobi method and a multigrid solver using the functions available in AI libraries. For the latter, we use a U-Net architecture which is able to represent a sawtooth multigrid method. A benefit of using AI libraries in this way is that one can exploit their power and their built-in technologies. For example, their executions are already optimised for different computer architectures, whether it be CPUs, GPUs or new-generation AI processors. In this article, we apply the proposed approach to eigenvalue problems in reactor physics where neutron transport is described by diffusion theory. For a fuel assembly benchmark, we demonstrate that the solution obtained from our new approach is the same (accounting for solver tolerances) as that obtained from the same discretisation coded in a standard way using Fortran. We then proceed to solve a reactor core benchmark using the new approach.

C. J. Cotter, D. A. Ham, C. C. Pain et al.

We introduce a new mixed discontinuous/continuous Galerkin finite element for solving the 2- and 3-dimensional wave equations and equations of incompressible flow. The element, which we refer to as P1dg-P2, uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure. The aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the LBB stability condition and hence has no spurious zero-energy modes. We illustrate this property with numerical integrations of the wave equation in two dimensions, an analysis of the resultant discrete Laplace operator in two and three dimensions, and a normal mode analysis of the semi-discrete wave equation in one dimension.

T. R. F. Phillips, C. E. Heaney, C. Boyang et al.

In this paper we solve the Boltzmann transport equation using AI libraries. The reason why this is attractive is because it enables one to use the highly optimised software within AI libraries, enabling one to run on different computer architectures and enables one to tap into the vast quantity of community based software that has been developed for AI and ML applications e.g. mixed arithmetic precision or model parallelism. Here we take the first steps towards developing this approach for the Boltzmann transport equation and develop the necessary methods in order to do that effectively. This includes: 1) A space-angle multigrid solution method that can extract the level of parallelism necessary to run efficiently on GPUs or new AI computers. 2) A new Convolutional Finite Element Method (ConvFEM) that greatly simplifies the implementation of high order finite elements (quadratic to quintic, say). 3) A new non-linear Petrov-Galerkin method that introduces dissipation anisotropically.

C. J. Cotter, D. A. Ham, C. C. Pain

We introduce a mixed discontinuous/continuous finite element pair for ocean modelling, with continuous quadratic pressure/layer depth and discontinuous velocity. We investigate the finite element pair applied to the linear shallow-water equations on an f-plane. The element pair has the property that all geostrophically balanced states which strongly satisfy the boundary conditions have discrete divergence equal to exactly zero and hence are exactly steady states of the discretised equations. This means that the finite element pair has excellent geostrophic balance properties. We illustrate these properties using numerical tests and provide convergence calculations which show that the discretisation has quadratic errors, indicating that the element pair is stable.

S. C. Kramer, C. J. Cotter, C. C. Pain

We discuss the ill conditioning of the matrix for the discretised Poisson equation in the small aspect ratio limit, and motivate this problem in the context of nonhydrostatic ocean modelling. Efficient iterative solvers for the Poisson equation in small aspect ratio domains are crucial for the successful development of nonhydrostatic ocean models on unstructured meshes. We introduce a new multigrid preconditioner for the Poisson problem which can be used with finite element discretisations on general unstructured meshes; this preconditioner is motivated by the fact that the Poisson problem has a condition number which is independent of aspect ratio when Dirichlet boundary conditions are imposed on the top surface of the domain. This leads to the first level in an algebraic multigrid solver (which can be extended by further conventional algebraic multigrid stages), and an additive smoother. We illustrate the method with numerical tests on unstructured meshes, which show that the preconditioner makes a dramatic improvement on a more standard multigrid preconditioner approach, and also show that the additive smoother produces better results than standard SOR smoothing. This new solver method makes it feasible to run nonhydrostatic unstructured mesh ocean models in small aspect ratio domains.