Exact solution and the multidimensional Godunov scheme for the acoustic equations

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

Modeling and Numerical Simulation of a Vegetation Cover

The aim of this work is to introduce a mathematical model representing the evolution of the temperature in a vegetation cover and the ground underneath it. Vegetation, and its interaction with soil, plays a very important role in the protection of soil surface from the action of sun and precipitations. A reduction in the vegetated mass increase the risk of desertification, soil erosion or surface runoffs which which can give rise to soil loss and sediment retention. These processes can favour climate change and global warming, which are major concerns nowadays. The mathematical model presented takes into account the main processes involved in vegetation cover and the interaction with the soil, among which, we can mention the Leaf Area Index, which is a dimensionless quantity defined as the one-sided green leaf area per unit ground surface area, or albedo and co-albedo which are clearly influenced by the vegetation. It is also considered a nonlinear heat capacity in the soil which incorporates the latent heat of fusion, when the phase change takes place. The numerical technique used to solve the mathematical model is based on a finite volume scheme with Weighted Essentially Non Oscillatory technique for spatial reconstruction and the third order Runge-Kutta Total Variation Diminishing numerical scheme is used for time integration. Some numerical examples are solved to obtain the distribution of temperature both in the vegetation cover and the soil.

Topography-based local spherical Voronoi grid refinement on classical and moist shallow-water finite-volume models

Locally refined grids for global atmospheric models are attractive since they are expected to provide an alternative to solve local phenomena without the requirement of a global high-resolution uniform grid, whose computational cost may be prohibitive. Spherical centroidal Voronoi tessellation (SCVT), as used in the atmospheric Model for Prediction Across Scales (MPAS), allows a flexible way to build and work with local refinement. In addition, the Andes Range plays a key role in the South American weather, but it is hard to capture its fine-structure dynamics in global models. This paper describes how to generate SCVT grids that are locally refined in South America and that also capture the sharp topography of the Andes Range by defining a density function based on topography and smoothing techniques. We investigate the use of the mimetic finite-volume scheme employed in the MPAS dynamical core on this grid considering the nonlinear classic and moist shallow-water equations on the sphere. We show that the local refinement, even with very smooth transitions from different resolutions, generates spurious numerical inertia–gravity waves that may even numerically destabilize the model. In the moist shallow-water model, wherein physical processes such as precipitation and cloud formation are included, our results show that the local refinement may generate spurious rain that is not observed in uniform-resolution SCVT grids. Fortunately, the spurious waves originate from small-scale grid-related numerical errors and can therefore be mitigated using fourth-order hyperdiffusion. We exploit a grid geometry-based hyperdiffusion that is able to stabilize spurious waves and has very little impact on the total energy conservation. We show that, in some cases, the clouds are better represented in a variable-resolution grid when compared to a respective uniform-resolution grid with the same number of cells, while in other cases, grid effects can affect the cloud and rain representation.

A Collocated Finite Volume Scheme for High-Performance Simulation of Induced Seismicity in Geo-Energy Applications

We develop a collocated Finite Volume Method (FVM) to study induced seismicity as a result of pore pressure fluctuations. A discrete system is obtained based on a fully-implicit coupled description of flow, elastic deformation, and contact mechanics at fault surfaces on a fully unstructured mesh. The cell-centered collocated scheme leads to convenient integration of the different physical equations, as the unknowns share the same discrete locations on the mesh. Additionally, a multi-point flux approximation is formulated in a general procedure to treat heterogeneity, anisotropy, and cross-derivative terms for both flow and mechanics equations. The resulting system, though flexible and accurate, can lead to excessive computational costs for field-relevant applications. To resolve this limitation, a scalable parallel solution algorithm is developed and presented. Several proof-of-concept numerical tests, including benchmark studies with analytical solutions, are investigated. It is found that the presented method is indeed accurate, stable and efficient; and as such promising for accurate and efficient simulation of induced seismicity.