# Browsing by Subject "Conservation laws"

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Item A fragmentation model for sprays and L² stability estimates for shockes solutions of scalar conservation laws using the relative entropy method(2010-05) Leger, Nicholas Matthew; Vasseur, Alexis F.; Arbogast, Todd J.; Gamba, Irene M.; Vishik, Mikhail M.; Raman, VenkatramananShow more We present a mathematical study of two conservative systems in fluid mechanics. First, we study a fragmentation model for sprays. The model takes into account the break-up of spray droplets due to drag forces. In particular, we establish the existence of global weak solutions to a system of incompressible Navier-Stokes equations coupled with a Boltzmann-like kinetic equation. We assume the particles initially have bounded radii and bounded velocities relative to the gas, and we show that those bounds remain as the system evolves. One interesting feature of the model is the apparent accumulation of particles with arbitrarily small radii. As a result, there can be no nontrivial hydrodynamical equilibrium for this system. Next, with an interest in understanding hydrodynamical limits in discontinuous regimes, we study a classical model for shock waves. Specifically, we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L² perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L² norm of a perturbed solution relative to the shock wave is bounded above by the L² norm of the initial perturbation. Finally, we include some preliminary relative entropy estimates which are suitable for a study of shock wave solutions to n x n systems of conservation laws having a convex entropy.Show more Item Task-based parallelism for hurricane storm surge modeling(2020-07-30) Bremer, Maximilian Heimo Moritz; Dawson, Clinton N.; Biros, George; Gamba, Irene; Heimbach, Patrick; Pingali, KeshavShow more Hurricanes are incredibly devastating events, constituting seven of the ten most costly U.S. natural disasters since 1980. The development of real-time forecasting models that accurately capture a storm's dynamics play an essential role in informing local officials' emergency management decisions. ADCIRC is one such model that is operationally active in the National Oceanic and Atmospheric Administration's Hurricane Surge On-Demand Forecast System. However, ADCIRC faces several limitations. It struggles solving highly advective flows and is not locally mass conservative. These aspects limit applicable flow regimes and can cause unphysical behavior. One proposed alternative which addresses these limitations is the discontinuous Galerkin (DG) finite element method. However, the DG method's high computational cost makes it unsuitable for real-time forecasting and has limited adoption among coastal engineers. Simultaneously, efforts to build an exascale machine and the resulting power constrained computing architectures have led to massive increases in the concurrency applications are expected to manage. These architectural shifts have in turn caused some groups to turn away from the traditional flat MPI or MPI+OpenMP programming models to more functional task-based programming models, designed specifically to be performant on these next generation architectures. The aim of this thesis is to utilize these new task-based programming models to accelerate DG simulations for coastal applications. We explore two strategies for accelerating the DG method for storm surge simulation. The first strategy addresses load imbalance caused by coastal flooding. During the simulation of hurricane storm surge, cells are classified as either wet or dry. Dry cells can trivially update, while wet cells require full evaluation of the physics. As the storm makes landfall and causes flooding, this generates a load imbalance. We present two load balancing strategies---an asynchronous diffusion-based approach and semi-static approach---to optimize compute resource utilization. These load balancing strategies are analyzed using a discrete-event simulation that models the task-based storm surge simulation. We find speed-ups of up to 56% over the currently used mesh partitioning and up to 97% of the theoretical speed-up. The second strategy focuses on a first order adaptive local timestepping scheme for nonlinear conservation laws. For problems such as hurricane storm surge, the global CFL timestepping constraint is overly stringent for the majority of cells. We present a timestepping scheme that allows cells to stably advance based on local stability constraints. Since allowable timestep sizes depend on the state of the solution, care must be taken not to incur causality errors. The algorithm is accompanied with a proof of formal correctness that ensures that with a sufficiently small minimum timestep, the solution exhibits desired characteristics such as a maximum principle and total variation stability. The algorithm is parallelized using a speculative discrete event simulator. Performance results show that the implementation recovers 59%-77% of the optimal speed-up.Show more