The remapped particle-mesh advection scheme

We describe the remapped particle-mesh method, a new mass-conserving method for solving the density equation which is suitable for combining with semi-Lagrangian methods for compressible flow applied to numerical weather prediction. In addition to the conservation property, the remapped particle-mesh method is computationally efficient and at least as accurate as current semi-Lagrangian methods based on cubic interpolation. We provide results of tests of the method in the plane, results from incorporating the advection method into a semi-Lagrangian method for the rotating shallow-water equations in planar geometry, and results from extending the method to the surface of a sphere

An inherently mass-conserving semi-implicit semi-Lagrangian discretisation of the shallow-water equations on the sphere

Copyright © 2009 Royal Meteorological SocietyFor the shallow-water equations on the sphere, an inherently mass-conserving semi-Lagrangian discretisation (SLICE) of the continuity equation is coupled with a semi-implicit semi-Lagrangian discretisation of the momentum equations. Various tests from the literature (two with analytical nonlinear solutions) are used to assess the model's performance and also to compare it with that of a variant model that instead employs a standard non-conserving semi-implicit semi-Lagrangian discretisation of the continuity equation. The mass-conserving version gives results that are overall somewhat better than the non-conserving one

Numerical Computation of Moving Boundary Phenomena

When matter is subjected to a gradient of: temperature, pressure, concentration, voltage or chemical potential a phase change may occur, which for dynamic processes will be separated by moving boundaries between the adjacent phases. Transport properties vary considerably between phases, consequently any change in phase modifies the rate of transport of: energy, momentum, charge or matter which are fundamental to the behaviour of many physical systems. Such dynamic multi-phase problems have, for historical and mathematical reasons, become known as either: Stefan problems or Moving Boundary Problems (MBPs). In most engineering applications the analysis of these problems is often impossible without recourse to numerical schemes which utilise either: finite difference or finite element methods. The success of finite element methods is their ability to handle complex geometries; however, they are time consuming and less amenable to vectorisation than finite difference techniques which, because of their greater simplicity in formulation and programming, continue to be the more popular choice. Several finite difference schemes are available for the solution of moving boundary problems; however, there are some difficulties associated with each method. Each time a new numerical scheme is developed, it has the aim of improving either, or both, the accuracy and the computational performance. For solving one-dimensional moving boundary problems, the variable time step grid is the best approach in terms of simplicity and computational efficiency. Due to the fact that the time step is variable the implicit recurrence formulae, which are stable for any mesh size, have always been used with this type of discretisation of the space time domain. It will be shown in the course of this thesis that the implicit methods are very inaccurate when used with relatively large time steps; hence, the immediate conclusion may be made - that implicit variable time step methods may not be sufficiently accurate to solve moving boundary problems where the boundary is moving with a relatively slow velocity. The proposed idea, of combining real and virtual grid networks and using new explicit finite difference equations, eliminates the loss of accuracy associated with implicit methods, when the time step is large, and offers higher computational performance. The new finite difference equations are based on the approach of making the finite difference substitution into the solution of the partial differential equation rather than into the partial differential equation itself, which is the classical approach. A new numerical scheme for two-phase Stefan problems which will be referred to as the EVTS method is developed and the solution is compared to other numerical methods as well as the analytic solution. Furthermore, the EVTS method is modified to solve implicit moving boundary problems (oxygen diffusion problem), in which an explicit relation containing the velocity of the moving boundary is absent. The resulting method achieves similar results to other more complex and time consuming methods. A further numerical scheme referred to as the ZC method is developed to deal with heat transfer problems involving three phases (or 2 moving boundaries) which appear and disappear during the process. To the knowledge of the author, a finite difference method for such a problem does not exist. For validation, numerical results are compared with those of the conservative finite element method of Bonnerot and Jamet, which is the only other method available to solve two-moving boundary problems. Finally, a new finite difference solution for non-linear problems is developed and applied to laser heat treatment of materials. The numerical results are in good agreement with published experimental results

Multi-stage high order semi-Lagrangian schemes for incompressible flows in Cartesian geometries

Efficient transport algorithms are essential to the numerical resolution of incompressible fluid flow problems. Semi-Lagrangian methods are widely used in grid based methods to achieve this aim. The accuracy of the interpolation strategy then determines the properties of the scheme. We introduce a simple multi-stage procedure which can easily be used to increase the order of accuracy of a code based on multi-linear interpolations. This approach is an extension of a corrective algorithm introduced by Dupont \& Liu (2003, 2007). This multi-stage procedure can be easily implemented in existing parallel codes using a domain decomposition strategy, as the communications pattern is identical to that of the multi-linear scheme. We show how a combination of a forward and backward error correction can provide a third-order accurate scheme, thus significantly reducing diffusive effects while retaining a non-dispersive leading error term.Comment: 14 pages, 10 figure

A Forward semi-Lagrangian Method for the Numerical Solution of the Vlasov Equation

This work deals with the numerical solution of the Vlasov equation. This equation gives a kinetic description of the evolution of a plasma, and is coupled with Poisson's equation for the computation of the self-consistent electric field. The coupled model is non linear. A new semi-Lagrangian method, based on forward integration of the characteristics, is developed. The distribution function is updated on an eulerian grid, and the pseudo-particles located on the mesh's nodes follow the characteristics of the equation forward for one time step, and are deposited on the 16 nearest nodes. This is an explicit way of solving the Vlasov equation on a grid of the phase space, which makes it easier to develop high order time schemes than the backward method

Assessing implicit large eddy simulation for two‐dimensional flow

Numerical models of the atmosphere cannot resolve all relevant scales; the effects of unresolved scales on resolved scales must be represented by a subgrid model or parametrization. When the unresolved scales are similar in character to the resolved scales (as in three‐dimensional or layerwise two‐dimensional turbulence) the problem is essentially one of large eddy simulation. In this situation, one approach to subgrid modelling is implicit large eddy simulation (ILES), where the truncation errors of the numerical model attempt to act as the subgrid model. ILES has been shown to have some success for three‐dimensional turbulence, but the validity of the approach has not previously been examined for two‐dimensional or layerwise two‐dimensional flow, which is the regime relevant to weather and climate modelling. Two‐dimensional turbulence differs qualitatively from three‐dimensional turbulence in several ways, most notably in having upscale energy and downscale enstrophy transfers. The question is of practical importance since many atmospheric models in effect use the ILES approach, for example through the use of a semi‐Lagrangian advection scheme. In this paper a number of candidate numerical schemes are tested to determine whether their truncation errors can approximate the subgrid terms of the barotropic vorticity equation. Results show that some schemes can implicitly model the effects of the subgrid term associated with the stretching and thinning of vorticity filaments to unresolvable scales; the subgrid term is then diffusive and is associated with the downscale enstrophy transfer. Conservation of vorticity, by using a flux form scheme rather than advective form for advection of vorticity, was found to improve performance of a candidate ILES scheme. Some effects of the subgrid terms could not be captured by any of the schemes tested, whether using an implicit or a simple explicit subgrid model: none of the schemes tested is able to capture the upscale transfer of energy from unresolved to resolved scales. Copyright © 2011 Royal Meteorological Society and British Crown Copyright, the Met OfficePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90526/1/925_ftp.pd

A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106984/1/fld3887.pd

Simulations of idealised 3D atmospheric flows on terrestrial planets using LFRic-Atmosphere

We demonstrate that LFRic-Atmosphere, a model built using the Met Office's GungHo dynamical core, is able to reproduce idealised large-scale atmospheric circulation patterns specified by several widely-used benchmark recipes. This is motivated by the rapid rate of exoplanet discovery and the ever-growing need for numerical modelling and characterisation of their atmospheres. Here we present LFRic-Atmosphere's results for the idealised tests imitating circulation regimes commonly used in the exoplanet modelling community. The benchmarks include three analytic forcing cases: the standard Held-Suarez test, the Menou-Rauscher Earth-like test, and the Merlis-Schneider Tidally Locked Earth test. Qualitatively, LFRic-Atmosphere agrees well with other numerical models and shows excellent conservation properties in terms of total mass, angular momentum and kinetic energy. We then use LFRic-Atmosphere with a more realistic representation of physical processes (radiation, subgrid-scale mixing, convection, clouds) by configuring it for the four TRAPPIST-1 Habitable Atmosphere Intercomparison (THAI) scenarios. This is the first application of LFRic-Atmosphere to a possible climate of a confirmed terrestrial exoplanet. LFRic-Atmosphere reproduces the THAI scenarios within the spread of the existing models across a range of key climatic variables. Our work shows that LFRic-Atmosphere performs well in the seven benchmark tests for terrestrial atmospheres, justifying its use in future exoplanet climate studies.Comment: 34 pages, 9(12) figures; Submitted to Geoscientific Model Development; Comments are welcome (see Discussion tab on the journal's website: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-647

A generalised finite difference scheme based on compact integrated radial basis function for flow in heterogeneous soils

In the present paper, we develop a generalised finite difference approach based on compact integrated radial basis function (CIRBF) stencils for solving highly nonlinear Richards equation governing fluid movement in heterogeneous soils. The proposed CIRBF scheme enjoys a high level of accuracy and a fast convergence rate with grid refinement owing to the combination of the integrated RBF approximation and compact approximation where the spatial derivatives are discretised in terms of the information of neighbouring nodes in a stencil. The CIRBF method is first verified through the solution of ordinary differential equations, 2-D Poisson equations and a Taylor-Green vortex. Numerical comparisons show that the CIRBF method outperforms some other methods in the literature. The CIRBF method in conjunction with a rational function transformation method and an adaptive time-stepping scheme is then applied to simulate 1-D and 2-D soil infiltrations effectively. The proposed solutions are more accurate and converge faster than those of the finite different method employed with a second-order central difference scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the 1D-IRBF and HOC schemes