Affordable, Entropy Conserving and Entropy Stable Flux Functions for the Ideal MHD Equations

In this work, we design an entropy stable, finite volume approximation for the ideal magnetohydrodynamics (MHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that discretely preserves the entropy of the system. To guarantee the discrete conservation of entropy requires the addition of a particular source term to the ideal MHD system. Exact entropy conserving schemes cannot dissipate energy at shocks, thus to compute accurate solutions to problems that may develop shocks, we determine a dissipation term to guarantee entropy stability for the numerical scheme. Numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness.Comment: arXiv admin note: substantial text overlap with arXiv:1509.06902; text overlap with arXiv:1007.2606 by other author

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around $N=7$). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings

Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas

An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem

A Provably Stable Discontinuous Galerkin Spectral Element Approximation for Moving Hexahedral Meshes

We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free-stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results

Ambient vibration analysis of an unstable mountain slope

A field experiment with small aperture seismic arrays was performed on the unstable rock slope above the village of Randa in the southern Swiss Alps. The aim of this experiment was to constrain the seismic response of a potential future rockslide using ambient vibration recordings. Weak seismic events were identified on the recordings and site-to-reference spectral ratios were calculated using a reference site located on the stable part of the slope. Spectral ratios of up to 30 were observed at sites located within the unstable portion of the slope. A strong variation of spectral ratios with azimuth indicates a directional site effect. Neither amplification nor directionality were observed at sites located in the stable part of the slope. Furthermore, time-frequency polarization analysis of the ambient noise was performed to provide robust estimates of frequency dependent directions of the maximum polarization. It was found that the unstable part of the slope vibrates within a narrow range of directions (130 ± 10°) for the frequency range centred around 5 Hz. The polarization directions estimated from ambient seismic vibrations are in good agreement with the deformation directions obtained by geodetic and in situ measurements. No directionality of ambient vibrations was observed at sites within the stable part of the slop

Stocking strategies for a pre-alpine whitefish population under temperature stress

Cold-water fish stocks are increasingly affected by steadily increasing water temperatures. The question arises whether stock management can be adapted to mitigate the consequences of this climatic change. Here, we estimate the effects of increasing water temperatures on fisheries yield and population dynamics of whitefish, a typical cold-water fish species. Using a process-based population model calibrated on an empirical long-term data set for the whitefish population (Coregons lavaretus (L.) species complex) of the pre-alpine Lake Irrsee, Austria, we project density-dependent and temperature-dependent population growth and compare established stock enhancement strategies to alternative stocking strategies under the aspect of increasing water temperatures and cost neutrality. Additionally, we contrast the results obtained from the process-based model to the results from simple regression models and argue that the latter show qualitative inadequacies in projecting catch with rising temperatures. Our results indicate that increasing water temperatures reduce population biomass between 2.6% and 7.9% and catch by the fishery between 24% and 48%, depending on temperature scenario and natural mortality calculation. These reductions are caused by accelerated growth, smaller asymptotic size and lower annual survival of whitefish. Regarding stocking strategies under constant temperatures, we find that stocking mostly whitefish larvae, produces higher population biomass than stocking mostly one-summer-old whitefish, while catch remains almost constant. With increasing temperatures, stocking one-summer-old fish is more beneficial for the angling fishery. Adaption to climate change by changing stocking strategies cannot, however, prevent an overall reduction in catch and population size of this cold-water fish species

Freshwater Ecosystems: From Models to Applications

Freshwater ecosystems—lakes and streams—are being endangered by agricultural, urban, and industrial pollution; hydraulic engineering; and overexploitation, which threaten their capacity to provide important services (recreation and supply of food and clean water, among others). Ecological modeling may be employed to estimate impacts and analyze mitigation strategies. Toy models are easy to construct, but applying them to real-world problems is often challenging. Here, we show in two case studies how the connection from model to application can be made. The first study analyzes whether and how the impact of climatic change on a mostly recreational fishery in an Alpine lake can be mitigated, while the second looks at restoring biodiversity after cleaning up pollution in a Korean river system, using aquatic insects, which play an essential functional role in aquatic food-webs and are very sensitive to water quality, as indicators of ecosystem health. These studies highlight the ability of process-based eco-evolutionary models to generate testable hypotheses and contribute solutions to real-world problems