A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

An <i>N</i>-dimensional Fortran interpolation programme (NterGeo.v2020a) for geophysics sciences – application to a back-trajectory programme (Backplumes.v2020r1) using CHIMERE or WRF outputs

An interpolation programme coded in Fortran for irregular N-dimensional cases is presented and freely available. The need for interpolation procedures over irregular meshes or matrixes with interdependent input data dimensions is frequent in geophysical models. Also, these models often embed look-up tables of physics or chemistry modules. Fortran is a fast and powerful language and is highly portable. It is easy to interface models written in Fortran with each other. Our programme does not need any libraries; it is written in standard Fortran and tested with two usual compilers. The programme is fast and competitive compared to current Python libraries. A normalization option parameter is provided when considering different types of units on each dimension. Some tests and examples are provided and available in the code package. Moreover, a geophysical application embedding this interpolation programme is provided and discussed; it consists in determining back trajectories using chemistry-transport or mesoscale meteorological model outputs, respectively, from the widely used CHIMERE and Weather Research and Forecasting (WRF) models.

Global tide simulations with ICON-O: testing the model performance on highly irregular meshes

The global tide is simulated with the global ocean general circulation model ICON-O using a newly developed tidal module, which computes the full tidal potential. The simulated coastal M2 amplitudes, derived by a discrete Fourier transformation of the output sea level time series, are compared with the according values derived from satellite altimetry (TPXO-8 atlas). The experiments are repeated with four uniform and sixteen irregular triangular grids. The results show that the quality of the coastal tide simulation depends primarily on the coastal resolution and that the ocean interior can be resolved up to twenty times lower without causing considerable reductions in quality. The mesh transition zones between areas of different resolutions are formed by cell bisection and subsequent local spring optimisation tolerating a triangular cell’s maximum angle up to 84°. Numerical problems with these high-grade non-equiangular cells were not encountered. The results emphasise the numerical feasibility and potential efficiency of highly irregular computational meshes used by ICON-O.

Short communication: Landlab v2.0: a software package for Earth surface dynamics

Numerical simulation of the form and characteristics of Earth's surface provides insight into its evolution. Landlab is an open-source Python package that contains modularized elements of numerical models for Earth's surface, thus reducing time required for researchers to create new or reimplement existing models. Landlab contains a gridding engine which represents the model domain as a dual graph of structured quadrilaterals (e.g., raster) or irregular Voronoi polygon–Delaunay triangle mesh (e.g., regular hexagons, radially symmetric meshes, and fully irregular meshes). Landlab also contains components – modular implementations of single physical processes – and a suite of utilities that support numerical methods, input/output, and visualization. This contribution describes package development since version 1.0 and backward-compatibility-breaking changes that necessitate the new major release, version 2.0. Substantial changes include refactoring the grid, improving the component standard interface, dropping Python 2 support, and creating 31 new components – for a total of 58 components in the Landlab package. We describe reasons why many changes were made in order to provide insight for designers of future packages. We conclude by discussing lessons about the dynamics of scientific software development gained from the experience of using, developing, maintaining, and teaching with Landlab.

A N-dimensional Fortran Interpolation Program (NterGeo.v2020a) for Geophysics Sciences – Application to a back-trajectory program (BACKPLUMES.v2020r1) using CHIMERE or WRF outputs

An interpolation program coded in Fortran for irregular N-dimensional cases is presented and freely available. Needs of interpolation procedure over irregular meshes or matrixes with interdependent input data dimensions is frequent in geophysical models. Also, these models often embed look-up tables of physics/chemistry modules. Fortran is a powerful and fast language, highly portable and easy to interface with other existing Fortran models. Our program does not need any libraries and can be compiled with any Fortran compiler. The program is fast and competitive compared to current Python libraries. A novel optional parameter (normalisation option) is provided when considering different types of units on each dimension. For the general program, the inverse distance is used for the weight calculation with a distance defined as a p-distance. Some tests and examples are provided and available in the code package. Moreover, a real case of geophysics application embedding this interpolation program is provided and discussed, it consists in determining back-trajectories using atmospheric dispersion or mesoscale meteorological model outputs, respectively from the widely used models CHIMERE and WRF.

Positive Definite Advection Transport Algorithm for Conservation Law Equations on Nonuniform Irregular Grids

The multidimensional positive definite advection transport algorithm (MPDATA) is an important numerical method for the computation of atmospheric dynamics. MPDATA is second-order accurate, positive definite, conservative, and computationally efficient. However, the method is problematic in which it results in a loss of precision when computing a nonuniform irregular grid. Furthermore, research revealed two reasons for this problem. On the one hand, numerical discretization of boundary derivatives of the finite-volume method is incompatible with nonuniform meshes (or grids); on the other hand, the up-wind scheme of staggered grids is not applicable to the calculation of irregular grids. We overcome these two problems by using the multipoint Taylor expansion method to obtain a boundary derivative numerical approximation scheme that does not depend on the grid structure. Furthermore, combined with the well-balance central-upwind scheme, a positive definite advection scheme for irregular meshes is proposed. Then, the positivity of the new numerical scheme is analyzed. Finally, the result of this study is verified by numerical simulation.