Mathematics Home / 2017 Clifford Lectures

**This year's Clifford Lecturer: Randall J. LeVeque (University of Washington, Seattle)**

**Subjects**

The general theme of these lectures will be the development of software that can be used to model geophysical hazards using depth-averaged fluid dynamics equations, most suitable for flows that are shallow relative to their spatial extent. In particular, the nonlinear shallow water equations are often used for modeling tsunamis, storm surge, or overland flooding. The lectures will introduce the basic theory of hyperbolic PDEs, the development of high-resolution finite volume methods for their numerical solution, and the use of adaptive mesh refinement to concentrate computational effort where it is most needed. These methods can be applied to a wide variety of wave propagation problems beyond geohazards and have been implemented in the open source software package Clawpack. Modeling flows over topography introduces some new complications that led to the development of the GeoClaw software, as will be described in the final lecture.

**LECTURE 1: Hyperbolic Equations and Riemann Solvers**

Hyperbolic partial differential equations arise in many fields as models for wave propagation. Small amplitude waves can be modeled with linear hyperbolic systems (e.g. acoustics, linear elasticity) while nonlinear hyperbolic equations arise in compressible gas dynamics and modeling flow over topography, for example. Shock waves that arise in nonlinear problems can be challenging to compute with numerical methods, but a variety of effective methods have been developed that are based on solving Riemann problems at grid cell edges. The Riemann problem arising from piecewise constant initial data is also a key component in the mathematical theory of hyperbolic systems. This lecture will contain a summary of this theory and examples from several applications, and will feature demonstrations of Riemann solvers from a collection of Jupyter notebooks being developed in joint work with David Ketcheson and Mauricio del Razo; see https://github.com/clawpack/riemann_book for some examples.

**LECTURE 2: Finite Volume Methods and Adaptive Refinement**

Riemann solvers can be used to define finite volume methods, starting from Godunov's method as the simplest first-order accurate method and extending to higher order methods that still capture shocks sharply and robustly. A general class of methods will be described that can be applied to virtually any hyperbolic system for which an exact or approximate Riemann solver is available. For many problems it is essential to also use adaptive mesh refinement (AMR), starting with a coarse grid over a large domain and using several levels of refined grids to concentrate computational effort where it is most needed, for example only around a tsunami as it propagates across the ocean, with a much finer grid near a small coastal region where modeling of the resulting inundation is desired. General approaches to AMR will be discussed along with several examples from tsunami and storm surge modeling, and other applications.

**LECTURE 3: Clawpack Software and Open Source Development**

General finite volume methods for hyperbolic systems have been implemented in the Clawpack software (Conservation laws package), which has been under development since 1994 and includes adaptive mesh refinement in one, two, and three space dimensions. Early work by LeVeque and Marsha Berger was done without even the benefits of version control, but over the past decades the project has expanded to involve more than a dozen developers scattered around the world. This lecture will give an overview of some of the software engineering and community building tools that are now routinely used to coordinate this effort, and that has allowed extensions in new directions such as GeoClaw (for geophysical flows), D-Claw (David George's extension to debris flows and landslides), PyClaw (which includes higher order methods and MPI parallelization on uniform grids) and ForestClaw (Donna Calhoun's melding of Clawpack and GeoClaw with p4est, an oct-tree based AMR code that scales well in parallel). See http://www.clawpack.org.

**LECTURE 4: GeoHazard Modeling with GeoClaw**

Many geophysical flows have large spatial extent relative to the depth of the fluid, and so depth-averaged equations such as the shallow water equations (together with AMR) can be used to efficiently model these flows. A number of special difficulties arise in such applications, in particular the need to model the margins of the flow and the moving interface between wet and dry cells, and the need to work with real-world digital elevation models for the topography. For tsunami and storm surge simulations it is also necessary to model waves that have very small amplitude relative to the varying depth of the ocean at rest, requiring a "well balanced" numerical method for accurate computation. For many geophysical flows the shallow water equations are insufficient and more complex depth-averaged equations must instead be used. Challenges and algorithmic solutions will be discussed in the context of some specific applications.

**Invited speakers include:**

Marsha Berger (New York University)

Donna Calhoun (Boise State University)

Alina Chertock (North Carolina State University)

David George (U.S. Geological Survey)

Jennifer L. Irish (Virginia Tech)

Pushkar Kumar Jain (University of Texas at Austin)

Alexander Kurganov (Tulane University & Southern University of Science and Technology of China)

Kyle T. Mandli (Columbia University)

Talea L. Mayo (University of Central Florida)

Majid Mohammadian (University of Ottawa)

Jacques Sainte-Marie (Inria Paris-Rocquencourt)

**Title: GeoClaw Tutorial: Introduction to using the software for modeling tsunamis and storm surge**

GeoClaw (http://www.geoclaw.org) is an open-source software package for solving two-dimensional depth-averaged equations over general topography using high-resolution finite volume methods and adaptive mesh refinement. Wetting-and-drying algorithms allow modeling inundation or overland flows. The primary applications where GeoClaw has been used are tsunami modeling and storm surge, although it has also been applied to dam break floods and it forms the basis for the debris flow and landslide code D-Claw under development at the USGS Cascades Volcano Observatory.

This tutorial will give an introduction to setting up a tsunami or storm surge modeling problem in GeoClaw, including:

- Overview of capabilities,

- Installing the software,

- Using Python tools provided in GeoClaw to acquire and work with

topography DEMs and other datasets,

- Setting run-time parameters, including specifying adaptive refinement

regions,

- Options to output snapshots of the solution or maximum flow depths,

arrival times, etc.

- The VisClaw plotting software to visualize results using Python tools or

display on Google Earth.

GeoClaw is distributed as part of Clawpack (http://www.clawpack.org). If you wish to follow along, it is recommended that you install the software in advance on your laptop, please see http://www.clawpack.org/installing.html.

**Title: Modeling and simulation of asteroid-generated tsunamis**

Four years ago, an asteroid with a 20 meter diameter exploded in the atmosphere over Chelyabinsk, Russia, causing serious damage 20 kilometers away but no deaths. In 1908 an asteroid two to three times larger exploded in the atmosphere over Tunguska, Russia. The resulting blast wave leveled approximately 2000 square kilometers of forest. We are studying the question of what would occur if such an air burst happened over the ocean. Would the blast wave generate a tsunami that could threaten coastal cities far away?

In this talk we present simulations using GeoClaw for tsunami propagation from asteroid-generated air bursts under a range of conditions and bathymetry. We also use a one dimensional model problem based on the shallow water equations that has an explicit solution in closed form to understand some of the results.

However, airburst-generated waves are qualitatively different than earthquake generated tsunamis. The shorter length scales of the airburst may make the shallow water equation model (which is a long wave approximation) inaccurate. The short time scales may make water compressibility important. We extend the model problem to the linearized Euler equations to explore the effects of dispersion and compressibility. We end with a discussion of whether or not the shallow water equations are an appropriate model for this kind of tsunami.

**Title: Geo-ForestClaw : Modeling dam break simulations using scalable, spatially adaptive quad-trees**

We demonstrate our success in incorporating GeoClaw (LeVeque, George, Berger, Mandli), a widely used code for simulation tsunamis, debris flow, flooding, storm surges and so on, into ForestClaw, an adaptive quadtree code based on the highly scalable library p4est (C. Burstedde). This new adaptive mesh framework allows us to run GeoClaw simulations on large scale parallel computing environments, and achieve resolutions not available on a desktop computers. We will demonstrate results from recent tsumami events, as well as an historical dam break problem.

**Title: Structure Preserving Numerical Methods for Shallow Water and Related Models**

Shallow water and related models are widely used as a mathematical framework to study water flows in rivers and coastal areas as well as to investigate a variety of phenomena in atmospheric sciences and oceanography. These models are systems of time-dependent PDEs that are derived using physical properties such as conservation of mass and momentum, and hydrostatic or barotropic approximations.

Solving shallow water systems numerically is a challenging task because of several reasons. First, many physically relevant solutions of shallow water systems are small perturbations of steady states, characterized by a delicate balance between the flux and source terms. If the method does not accurately respect this balance, the numerical errors (which cannot be made too small on practically relevant grids) may lead to oscillations, in which the magnitude of artificial waves may be larger than the magnitude of the solution itself. The second major difficulty is related to the computation of solutions when the water depth is very small or even zero. In such a case, small numerical oscillations may lead to appearance of negative values of water depth, which are physically irrelevant.

In this talk, I will discuss several shallow water models (in particular those with friction and Coriolis terms) and present semi-discrete second-order central-upwind schemes that are capable of exactly preserving physically relevant steady states and maintaining the positivity of the water depth.

**Title: Simulating shallow earth-surface flows with a two-phase granular-fluid model.**

A large class of geophysical flows feature mixtures of both solid granular materials and water. Examples include landslides and debris flows, floods that entrain solid geological material, tsunamis generated by landslides, and tsunami inundation that entrains debris. For these problems, we use a two-phase depth averaged model, similar to the shallow water equations. The model was developed to simulate landslide failure and runout, and incorporates principles of solid, fluid and soil mechanics. Because the model is two-phase, it can also be used to model fluid problems in the absence of solid content (the model equations reduce to the shallow water equations with vanishing solid phase). We have recently extended the model to problems that involve granular flows that interact with bodies of water (landslide generated tsunamis, dam outbreak floods that entrain debris and landslides that entrain water). I will describe this model and software and show some simulations of these problems.

**Title: Hurricane Surge Hazard Assessment**

Since 2005, the US experienced some of its largest surges and hurricane-related damage on record. These disasters highlight the critical need for a robust and accurate hurricane surge hazard assessment approach to support future disaster resilience and planning activities. Here, dimensionless scaling and hydrodynamics arguments are used to quantify the influence of hurricane variables and geographic characteristics of the surge response. The use of this physical scaling to develop surge response functions (SRF) enables fast algebraic calculation of maximum surge height at any location of interest for any hurricane meteorological condition, without loss of accuracy gained via high-resolution computational surge simulation. When coupled with joint probability statistics, the use of SRFs facilitates rapid development of continuous probability density functions for probabilistic hazard assessment. Considerations for determining the minimum set of discrete high-fidelity storm surge simulations required for accurate probabilistic hazard assessment will be presented. Applications of the hazard assessment approach to address future changes in hurricane climatology, sea level rise, and land-cover change will also be discussed.

**Title: Adaptive Moving Mesh Central-Upwind Schemes for the Saint-Venant Systems of Shallow Water Equations**

It is well-known that solutions of the Saint-Venant system of shallow water equations may be nonsmooth and thus they are understood in the weak sense. Therefore, finite-volume methods, which are based on integral formulation of hyperbolic system of PDEs, are appropriate numerical tool for computing this type of solutions. In the framework of finite-volume Godunov-type schemes, the solution, which is assumed to be available at a certain time level, is first reconstructed (approximated) globally in space using a piecewise polynomial function and then evolved in time to the next time level by solving the studied hyperbolic system subject to piecewise polynomial initial data. The evolution step is performed by integrating the system over a space-time control volume. Depending on the way the control volumes are selected, Godunov-type schemes can be split into two classes: upwind and central. To design upwind schemes, the numerical fluxes along the boundaries of the control volumes are to be computed by solving (generalized) Riemann problems, arising at each cell interface at the reconstruction step. Central schemes are much simpler since in the central framework, the fluxes are evaluated away from discontinuities and therefore no (generalized) Riemann problems need to be solved. Therefore, central schemes can be applied as a black-box solver to a variety of hyperbolic systems. However, compared to their upwind counterparts, central schemes contain larger amount of numerical diffusion, which may oversmear nonsmooth solutions. To reduce the numerical diffusion present in central schemes, we have developed central-upwind schemes, which I will present in the first part of my talk.

Application of the central-upwind to the Saint-Venant system requires development of several techniques, which will guarantee that a delicate balance between the flux and source terms is respected and that the positivity of the computed water depth is not disrupted. To achieve these goals we have developed a well-balanced positivity preserving central-upwind schemes, which I will present in the second part of my talk.

In the last part of my talk, I will show how one can achieve even higher resolution as well as to improve the efficiency of central-upwind schemes by developing new adaptive moving mesh (AMM) central-upwind schemes on both adaptive one-dimensional nonuniform grids and two-dimensional structured quadrilateral meshes. After evolving the solutions to the new time level, the grid points are redistributed according to the moving mesh PDE, and then the solution is projected onto the new mesh in a conservative manner. In order to preserve the positivity of the computed water depth, several measures are taken. First, we either make sure that the reconstructed values of the water surface stay above the corresponding values of the bottom topography or use the positivity preserving reconstructions for the water depth. Second, we use a draining time-step technique to ensure that the water depth remains positive during the evolution step. Third, we propose special corrections of the solution projection step in (almost) dry areas. I will demonstrate the ability of the proposed AMM central-upwind schemes to significantly improve quality of the computed solution in a number of numerical examples.

**Title: Computational Methods for Storm Surge**

Coastal flooding due to severe storms is one of the most wide-spread and damaging hazards faced around the world. The threat of these events has grown not only due to increased population and economic reliance on coastal regions but also due to climate change impacts such as sea-level rise. Computational predictive capabilities are critical to addressing this threat but require the ability to handle multiple, disparate scales, handle the physics relevant at each of these scales, and remain tractable under the necessity of large ensembles to handle uncertainty in the input. In this talk a number of efforts to addressing these and other issues related to storm surge computational models will be discussed. These include the use of adaptive mesh refinement, extensions to the modeling equations (such as the multi-layer shallow water equations), strategies for tackling multi-scale aspects of the problem, and work towards increasing computational efficiency.

In recent decades, storm surge models have become increasingly accurate due to im-provements in numerical methods, advancements in high performance computing, and im-provements in the estimation of model parameters as well as their representation. Unfor-tunately, uncertainties in storm surge models remain, and there is thus a need to quantify, and then reduce them. Specically, when storm surges are estimated in real-time, many uncertainties are introduced due to the uncertainties in the hurricane (i.e. wind) forecast itself. To aid in emergency response, an ensemble of hurricane scenarios is often used to determine a range of possibilities of storm surge inundation. However, the likelihood of each scenario remains unclear. Here, we discuss methods of estimating these likelihoods given the uncertainties in various storm parameters. We also investigate methods of de-termining the evolution of the likelihoods as the uncertainty in the hurricane forecasts is reduced.

**Title: Numerical simulation of shallow water flows with sediment transport and variable density**

Environmental free surface shallow water flows have numerous applications in transport of debris and suspended sediment especially over a sloped bed. When such flows enter larger rivers or other types of water bodies such as lakes, significant difference between the densities of the two systems need to be accounted for in order to ensure accurate simulation results. As such, domestic and industrial effluents from outfall structures often have a different density than that of the ambient water body, something which leads to various flow and mixing characteristics of the discharge.

To model the mixing of the two interacting water bodies, if justifiable, it is more efficient to use two-dimensional depth-averaged models because of their simplicity in implementation and application, especially in the initial stages of the design. Therefore, in many cases, if the ambient stream can be approximated to a shallow stream, the use of shallow water equations can lead to some of the most performant tools in modeling mixing problems. To use the traditional shallow water equations in modeling mixing, it is necessary to further modify them to include density change, which has been the focus of many previous studies using shallow water equations.

In this study, the central upwind scheme developed by Bryson et al. (2010) is applied and extended to variable density shallow water equations following the work of Brice et al. (2010). Due to the use of the Bousinesq approximation in the formulation suggested by Brice et al. (2010), it is assumed that the density gradients are small. Therefore, the methodology used here is more efficient compared to similar works by other investigators, when the concentration of the sediment in suspension is not very high. This scheme considers the well-balanced and positivity-preserving characteristics of the dense flows. In this regard, the analytical solution is presented over a triangular grid, using a high order temporal and spatial numerical scheme which is discussed in detail.

**Title: Free Surface Flows: from Hydrostatic to Non-Hydrostatic Models**

The equations governing free surface flows, typically the Navier-Stokes equations, are difficult to analyse and solve and therefore reduced complexity models are often used to represent geophysical flows. During this presentation, we first present models able to approximate the hydrostatic Navier-Stokes equations. The associated numerical schemes are endowed with stability properties such as positivity, well-balancing and discrete entropy inequality. They are confronted with analytical solutions and experimental measurements. Then we propose non-hydrostatic (dispersive) models and numerical procedures to approximate them. But the numerical analysis of such models is complex and several questions remain opened.

Abu Chackalamannil Thomas (Tulane University)

Alexander Hoover (Tulane University)

Amy Buchmann (Tulane University)

Forest Mannan (Tulane University)

John R. Lagrone (Tulane University)

Kun Zhao (Tulane University)

Li Guan (Tulane University)

Lin Li (Tulane University)

Lisa Fauci (Tulane University)

Mac Hyman (Tulane University)

Nathan E Glatt-Holtz (Tulane University)

Nick Cogan (Florida State University)

Onur Danaci (Tulane University)

Padi Fuster (Tulane University)

Pengfei Li (Tulane University)

Pushkar Kumar Jain (University of Texas at Austin)

Ricardo Cortez (Tulane University)

Roseanna Gossmann (Tulane University)

Sankhaneel Bisui (Tulane University)

Tewodros Amdeberhan (Tulane University)

Tong Wu (North Carolina State University)

Victor Moll (Tulane University)

Vincent Martinez (Tulane University)

Xiao Guan (Tulane University)

Zhe Qu (Tulane University)

Zhuolin Qu (Tulane University)