How we compute
Numerical methods
Spectral/hp element methods combine the geometric flexibility of finite elements with the accuracy of spectral methods. We work on the discretisations themselves: how to make them stable, how to place resolution where it is needed, and how to couple parts of a simulation that do not line up.
Most engineering simulation uses low-order methods, where accuracy is improved by adding more elements to the mesh. High-order methods instead offer an alternative route of enrichment: we raise the polynomial degree within each element, so that accuracy improves far faster for the same number of degrees of freedom. For the unsteady, vortex-dominated flows we simulate, that difference is what makes a simulation far more affordable: low-order schemes damp the very structures being studied.
The catch is that high-order methods are less forgiving: aliasing errors (that a dissipative scheme would quietly absorb) can destabilise a simulation entirely; boundary conditions that seem innocuous can degrade the order of accuracy; and the choice between continuous and discontinuous formulations changes the computational cost of the whole simulation. A good deal of our work is on these questions of stability and formulation, including comparisons that establish where each approach is actually the right one rather than assuming it.
Two threads are particularly active within our group. Adaptivity asks where the resolution should go: whether to refine the mesh, raise the polynomial order, or move the points that are already there, and how to decide automatically. Non-conformal interfaces deal with the case where two parts of a simulation meet without matching up, which is what happens whenever geometry slides or rotates relative to the rest of the domain, as in turbomachinery. Getting the flux across such an interface right, without losing accuracy or conservation, is a surprisingly delicate problem and one we have returned to repeatedly.
These methods are implemented in Nektar++, which the group leads.
arXiv preprint 2504.03573
Published version, CC BY, fig. 15
Funded by
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RAEng Research Chair
A five-year chair developing the next generation of solvers for nuclear fusion modelling, following on from the NEPTUNE projects.
Read more -
APinTA
Parallel-in-time methods for exascale computing.
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Designing, analysing and implementing numerical methods and deploying them across industrial, biomedical and environmental applications.
Who works on this
All of the team
Prof. David Moxey
Group leader and project leader of the Nektar++ framework.
Previously
Dr. Jacques Xing
Parallel-in-time methods for high-order methods.
Now: Research Software Engineer, Department of Aeronautics, Imperial College London
Dr. Edward Laughton
Developing high-fidelity parallel CFD for problems involving moving geometries.
Now: Trainee Patent Attorney, J A Kemp
Recent papers
All publications-
Architecture-aware h-to-p optimisation: spectral/hp element operators for mixed-element meshes
Comput. Phys. Commun., vol. 330, pp. 110467
We extend earlier international efforts to optimise hexahedral-based spectral element methods on GPUs and vectorised CPUs to mixed element meshes additionally involving prismatic, pyramidic, and tetrahedral shapes using tensorial expansions. We demonstrate that common finite element operators (such as the mass and Helmholtz matrices) benefit from alternative implementation strategies depending on the element shape, choice of polynomial order, and system architecture in order to achieve optimal performance. In addition, we introduce a new approach/interpretation to efficiently evaluate more complex operations involving inner products with the derivative of the expansions as part of the integrand such as the stiffness matrix. This approach seeks to maximise operations using the collocation properties of the nodal tensorial expansion associated with classical quadrature rules. Our GPU performance tests demonstrate that the throughput of the Helmholtz operator on tetrahedral elements is at most 2.5 times slower than on hexahedral elements, despite tetrahedra having a factor of six greater floating-point operations.@article{xing-2027, title = {Architecture-aware $h$-to-$p$ optimisation: spectral/$hp$ element operators for mixed-element meshes}, author = {Xing, J. Y. and Xia, B. and Renner, D. and Cantwell, C. D. and Moxey, D. and Kirby, R. M. and Sherwin, S. J.}, journal = {Comput. Phys. Commun.}, volume = {330}, pages = {110467}, year = {2027}, doi = {10.1016/j.cpc.2026.110467}, url = {https://www.sciencedirect.com/science/article/pii/S0010465526004480} } -
Comput. Phys. Commun., vol. 331, pp. 110473
Scale-resolving simulations of high-Reynolds-number incompressible flow are often limited by the Courant–Friedrichs–Lewy (CFL) restriction of explicit time-stepping schemes, leading to small steps and long times-to-solution. We systematically compare two implicit formulations of the velocity correction scheme – a linear-implicit approach and a sub-stepping (semi-Lagrangian) method – with a standard semi-implicit formulation in a high-order spectral/hp element framework. The schemes are assessed in terms of stability limits, temporal accuracy, and computational performance for implicit large-eddy simulation of the Imperial Front Wing benchmark, a complex high Reynolds number geometry with curved surfaces that imposes strict CFL constraints. The sub-stepping scheme extends the stable time step size by a factor of 20, and the linear-implicit scheme by up to a factor of 100 when equal-order velocity and pressure spaces are used. While increasing the cost per time step, the linear-implicit scheme reduces the time-to-solution by up to a factor of 8.7 at this largest step, whereas the sub-stepping scheme reduces it by only 12%. With application targets of 1% for mean lift and 5% for mean drag, the fastest accepted linear-implicit case reduces time-to-solution by a factor of 1.53 at ten times the CFL-limited step. The main-plane separation bubble is already substantially shortened at this step, showing that force agreement can mask changes in transition-sensitive flow structure. The results quantify the trade-off between stability, accuracy, and computational cost for implicit velocity correction schemes on complex geometries and provide guidance for selecting time integration strategies in large-scale scale-resolving simulations.@article{wustenberg-2027, title = {Implicit velocity correction schemes for scale-resolving simulations of incompressible flow: Stability, accuracy, and performance}, journal = {Comput. Phys. Commun.}, volume = {331}, pages = {110473}, year = {2027}, doi = {10.1016/j.cpc.2026.110473}, url = {https://www.sciencedirect.com/science/article/pii/S0010465526004546}, author = {Wüstenberg, H. and Liosi, A. and Sherwin, S. J. and Peiró, J. and Moxey, D.} } -
Unified interface flux evaluation in a general discontinuous Galerkin spectral element framework
SIAM J. Sci. Comput., vol. 48
High-order discontinuous Galerkin spectral element methods (DGSEM) have received growing attention and development, especially in the regime of computational fluid dynamics in recent years. The inherent flexibility of the discontinuous Galerkin approach in handling non-conforming interfaces, such as those encountered in moving geometries or hp-refinement, presents a significant advantage for real-world simulations. Despite the well-established mathematical framework of DG methods, practical implementation challenges persist to boost performance and capability. Most previous studies only focus on certain choices of element shape or basis type in a structured mesh, although they have demonstrated the capability of DGSEM in complex flow simulations. This work discusses the low-cost and unified interface flux evaluation approaches for general spectral elements in unstructured meshes, alongside their implementations in the open-source spectral element framework, Nektar++. The initial motivation arises from the discretization of Helmholtz equations by the symmetric interior penalty method, in which the system matrix can easily become non-symmetric if the flux is not properly evaluated on non-conforming interfaces. We focus on the polynomial non-conforming case in this work but extending to the geometric non-conforming case is theoretically possible. Comparisons of different approaches, trade-offs, and performance of benchmark of our initial matrix-free implementation are also included, contributing to the broader discourse on high-performance spectral element method implementations.@article{xia-2026, title = {Unified interface flux evaluation in a general discontinuous Galerkin spectral element framework}, author = {Xia, B. and Moxey, D.}, journal = {SIAM J. Sci. Comput.}, year = {2026}, doi = {10.1137/25M174824X}, volume = {48}, number = {3}, url = {https://doi.org/10.1137/25M174824X} } -
High-Fidelity Sliding Mesh Method for Moving Geometries in Turbomachinery Applications
Int. J. Turbomach. Propuls. Power, vol. 11
The accurate, high-fidelity modeling of moving geometries involving translation and rotation is important in fluid mechanics applications, particularly turbomachinery. These simulations facilitate the analysis of various turbomachinery stages, where complex movements are integral to their functionality. We integrated a sliding mesh capability into the iLES/DNS compressible Navier–Stokes solver within Nektar++, which is a spectral/hp element open-source framework. Static and moving regions of the finite element mesh are coupled across a non-conformal interface using a point-to-point interpolation method (PTPM). The solver is verified on two canonical benchmarks, the isentropic vortex and Couette flow, and validated on the T106A low-pressure turbine cascade with an upstream translating bar that generates inflow wakes, emulating rotor–stator interaction. This paper aims to provide robust validation of the new solver developments and supply a tool to understand how the upstream bar motion influences the downstream blade efficiency, which is crucial for enhancing turbomachinery design and performance.@article{ye-2026, author = {Ye, Junjie and Vivarelli, Guglielmo and Isler, João and Moxey, David and Sherwin, Spencer and Cantwell, Chris}, title = {High-Fidelity Sliding Mesh Method for Moving Geometries in Turbomachinery Applications}, journal = {Int. J. Turbomach. Propuls. Power}, volume = {11}, year = {2026}, number = {4}, url = {https://www.mdpi.com/2504-186X/11/4/40}, doi = {10.3390/ijtpp11040040} } -
J. Comput. Phys., vol. 524, pp. 113734
The high-fidelity modelling of geometry which rotates or translates is a key requirement in fluid mechanics applications, enabling the simulation of parts such as rotors and pressure cascades. These prescribed motions can be imposed through movement of the mesh representing the problem, where an interface is constructed across the non-conformal interface bridging the static and moving regions of the mesh, and the challenge is in maintaining solution accuracy of the numerical method across this interface. In this work, we consider the application of a point-to-point interpolation technique leveraging the high-order discontinuous Galerkin method, which has the advantage of being capable of handling complex interfaces that might not be possible under other interpolation techniques. We consider the use of this method within linear and non-linear hyperbolic systems, including the compressible Euler and Navier-Stokes equations, including a detailed analysis of observed error for canonical problems such as flow past a rotating cylinder.@article{lahooti-2025, title = {Discontinuous Galerkin simulation of sliding geometries using a point-to-point interpolation technique}, author = {Lahooti, M. and Laughton, E. and Ye, J. and Cantwell, C. D. and Moxey, D.}, journal = {J. Comput. Phys.}, volume = {524}, pages = {113734}, year = {2025}, doi = {10.1016/j.jcp.2025.113734}, url = {https://www.sciencedirect.com/science/article/pii/S0021999125000178} }
Collaborators
- Spencer Sherwin Imperial College London
- Joaquim Peiró Imperial College London
- Chris Cantwell Imperial College London
- Mike Kirby University of Utah
- Gianmarco Mengaldo National University of Singapore
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