EPrint Collection - Applied Mathematics

Permanent URI for this collectionhttps://digital.lib.washington.edu/handle/1773/1977

The EPrint Collection is a permanent and freely available archive of preprints and postprints of work by faculty and students associated with the Department of Applied Mathematics at the University of Washington.

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Accelerating invasions and the asymptotics of fat-tailed dispersal
(2019) Liu, B. R.; Kot, M.
Integrodifference equations (IDEs) are used in ecology to model the growth and spatial spread of populations. With IDEs, dispersal is specified with a probability density function, called the dispersal kernel, and the shape of the kernel influences how rapidly invasions progress. In this paper, we apply tail additivity, a property of regularly varying probability densities, to model invasions with fat-tailed (power-law decay) dispersal in one dimension. We show that fat- tailed invasions progress geometrically fast, with the rate of spread depending on the degree of fatness of the tails. Our analyses apply to populations with no Allee effect as well as weak Allee effects, and we conduct simulations to show that fat-tailed invasions with weak Allee effects produce accelerating invasions. We analyze point-release and front-release invasions, corresponding to newly-established and well-established populations, and find that front-release invasions gain a permanent speed-up over point-release invasions, invading at a faster geometric rate that persists for all time. Since accelerating invasions are qualitatively different than constant-speed invasions, we also discuss how measures of invasion must be modified and reconsidered when invasions accelerate.
Probabilistic Source Selection for the Cascadia Subduction Zone
(2017-03-19) Adams, Loyce; LeVeque, Randall J; Rim, Donsub; Gonzalez, Frank I
This report has been submitted to FEMA Region IX as a final project report for a project on developing new methodologies for Probabilistic Tsunami Hazard Assessment (PTHA). We propose a methodology for taking a large number of realizations of potential future earthquakes (with associated probabilities) and producing good approximations to the resulting hazard curves and maps without doing a computationally- expensive fine-grid tsunami simulation for each realization.
Tsunami Hazard Assessment of Whatcom County, Washington. Project Report - Version 2
(2019-05-19) Adams, Loyce; LeVeque, Randall J; Gonzalez, Frank
This report documents the results of a study supported by the Washington State Emergency Management Division of the tsunami hazard along the coast of Whatcom County. One earthquake source from the Seattle Fault and one from the Cascadia Subduction Zone were considered. Results include inundation depths and times of arrival that will be useful to coastal communities, as well as tsunami current speeds and momentum flux. GeoClaw Version 5.5.0 was used for the modeling, with some modifications as described in the appendices.
Issues Encountered with ASCE Compatibility Criteria
(2019-05) Adams, Loyce; Gonzalez, Frank; LeVeque, Randall J
The State of Washington is assisting at-risk coastal communities that have included the design and construction of tsunami vertical evacuation structures in their hazard mitigation plans. Washington State has not formally adopted the ASCE 7-16 Chapter 6 standard; however, past VES projects have made the decision to meet these standards, as well as TLES-approved ASCE 7 Change Proposals to revise Chapter 6 of the 2022 version, ASCE 7-22 (e.g., Chock, et al., 2018) in anticipation of possible formal approval by the ASCE 7 Main Committee and adoption by the State. As a result, Washington State has been a very active user of the standard, which continues to evolve as the TLES reviews and develops and votes on 7-22 Change Proposals. The purpose of this brief report is to contribute to the ASCE 7 TLES process of improving ASCE Chapter 6 guidance by identifying issues we have encountered with these standards and providing appropriate suggestions that we hope will improve the guidance and ease of use by practitioners.
Developing a Warning System for Inbound Tsunamis from the Cascadia Subduction Zone
(2018-01-07) LeVeque, Randall J; Bodin, Paul; Cram, Geoffrey; Crowell, Brendan W.; Gonzalez, Frank I.; Harrington, Michael; Mannalang, Dana; Melgar, Diego; Schmidt, David A.; Vidale, John E.; Vogl, Christopher J.; Wilcock, William S.
Real-time tsunami warning in the nearfield is considerably more difficult than producing warnings for distant events. Although in some cases strong shaking will provide the only warning, there are several situations in which better early tsunami warning systems could be critical. We discuss some of the issues that arise, particularly the difficulty of interpreting ocean bottom pressure recordings in the near source region, and make some recommendations for future research and first steps toward a better warning system for the Pacific Northwest.
GeoClaw Model Tsunamis Compared to Tide Gauge Results Final Report
(2018-11-03) Adams, Loyce M.; LeVeque, Randall J
The purpose of this project is to compare GeoClaw tsunami model results to detided tide gauge results at multiple destinations for each of several tsunamis. In particular, we are interested in the suitability of GeoClaw for calculating tsunami amplitudes with enough precision to be used for forecasting, especially in the context of ensemble modelling. In each of our comparison plots, we also include a sample MOST tsunami result which is useful to see the reasonableness of GeoClaw in regions where tide gauge data is missing or has insufficient resolution. The methodology behind GeoClaw can be found in [1] and [5], and its performance on the 2011 NTHMP problems in [4] and [6]. For a description of the MOST model see [7]. Here, we give a quick summary of our progress on such comparisons for the Japan 2011, Samoa 2009, Kuril 2007, Chile 2010 and HaidiGwaii 2012 tsunamis at tide gauge destinations at Crescent City, Arena Cove, Port Orford, Hilo, Midway Island and Pago Pago. In the next sections, we provide more details.
Tsunami Hazard Assessment of the Strait of Juan de Fuca
(2015-09-24) Gonzalez, Frank I.; LeVeque, Randall J; Adams, Loyce M.
This report documents the results of a study supported by the Washington State Emergency Management Division of the tsunami hazard along the Strait of Juan de Fuca. Results include inundation depths and times of arrival that will be useful to communities along the Strait as well as speeds, momentum, momentum flux, and minimum water depths that are useful for harbor masters and the major shipping and ferry industries that operate within the Strait of Juan de Fuca.
Probabilistic Tsunami Hazard Assessment (PTHA) for Crescent City, CA. Final Report for Phase I
(University of Washington Department of Applied Mathmatics, 2013-02-02) Gonzalez, Frank I.; LeVeque, Randall J; Adams, Loyce M.
This demonstration Probabilistic Tsunami Hazard Assessment (PTHA) study of Crescent City, California was funded by BakerAECOM and motivated by FEMA's desire to explore methods to improve products of the FEMA Risk Mapping, Assessment, and Planning (Risk MAP) Program. The primary results, that is, the 100- and 500-year tsunami maps, are presented and discussed in Section 3 and Appendix C. These maps were generated by a signi cantly improved methodology than that of the Seaside study; the improvements include a more complete set of seismic sources (Table 1), and a more accurate method for estimating tidal uncertainty (Section 8). As expected, the inland extent and magnitude of the ooding for the 500-year tsunami far exceed that of the 100-year event; these products can now be compared with standard FEMA Flood Insurance Rate Maps (FIRMs) to determine whether these ooding levels exceed estimates of other coastal ooding hazards, such as storm surge. A nal deliverable, digital data les of the map data, have been provided to BakerAECOM, and a description of these les is given in Appendix D. The primary conclusion we have reached in the course of this study is that the maps must be used with caution because (a) there are signi cant uncertainties in the speci ation of CSZ seismic sources (Section 4 and Section 7.6) and (b) the standard 100- and 500-year maps are highly sensitive to these geophysical uncertainties and, in certain circumstances could even be misleading. Section 9.2 discusses this sensitivity in the context of a non-regulatory product that provides valuable additional insight by presenting the same probabilistic information in a di erent format that we call a p-contour map. Finally, we recommend (a) that FEMA give serious consideration to the adoption of the p-contour map as a product that supplements and aids in the practical interpretation of the same probabilistic information displayed in the standard 100- and 500-year tsunami maps, and that (b) future PTHA studies should include close collaboration with a geoscientist expert in earthquake parameterization.
Finite volume methods and adaptive refinement for global tsunami propagation and local inundation
(2006) George, David L.; LeVeque, Randall J
The shallow water equations are a commonly accepted approximation governing tsunami propagation. Numerically capturing certain features of local tsunami inundation requires solving these equations in their physically relevant conservative form, as integral conservation laws for depth and momentum. This form of the equations presents challenges when trying to numerically model global tsunami propagation, so often the best numerical methods for the local inundation regime are not suitable for the global propagation regime. The different regimes of tsunami flow belong to different spatial scales as well, and require correspondingly different grid resolutions. The long wavelength of deep ocean tsunamis requires a large global scale computing domain, yet near the shore the propagating energy is compressed and focused by bathymetry in unpredictable ways. This can lead to large variations in energy and run-up even over small localized regions. We have developed a finite volume method to deal with the diverse flow regimes of tsunamis. These methods are well suited for the inundation regime—they are robust in the presence of bores and steep gradients, or drying regions, and can capture the inundating shoreline and run-up features. Additionally, these methods are well-balanced, meaning that they can appropriately model global propagation. To deal with the disparate spatial scales, we have used adaptive refinement algorithms originally developed for gas dynamics, where often steep variation is highly localized at a given time, but moves throughout the domain. These algorithms allow evolving Cartesian sub-grids that can move with the propagating waves and highly resolve local inundation of impacted areas in a single global scale computation. Because the dry regions are part of the computing domain, simple rectangular cartesian grids eliminate the need for complex shoreline-fitted mesh generation.
High-resolution finite volume methods for dusty gas jets and plumes
(2006) Pelanti, Marcia; LeVeque, Randall J
We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A high-resolution wave-propagation algorithm is used to solve the equations numerically. The one-dimensional algorithm is shown to give agreement with a shock tube test problem in the literature. The two-dimensional algorithm has been applied to model explosive volcanic eruptions in which an axisymmetric jet of hot dusty gas is injected into the atmospher and the expected behavior is observed at two different vent velocities. The methodology described here, with extensions to three dimensions and adaptive mesh refinement, is being used for more detailed studies of volcanic jet processes.
CORRECTION TO THE ARTICLE A COMPARISON OF THE EXTENDED FINITE ELEMENT METHOD WITH THE IMMERSED INTERFACE METHOD FOR ELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS AND SINGULAR SOURCES BY VAUGHAN ET AL.
(2008) Beale, J. T. (John Thomas), 1947-; Chopp, David L.; LeVeque, Randall J; Li, Zhilin, 1956-
A recent paper by Vaughan, Smith, and Chopp [Comm. App. Math. &and Comp. Sci. 1 (2006), 207–228] reported numerical results for three examples using the immersed interface method (IIM) and the extended finite element method (X-FEM). The results presented for the IIM showed first-order accuracy for the solution and inaccurate values of the normal derivative at the interface. This was due to an error in the implementation. The purpose of this note is to present correct results using the IIM for the same examples used in that paper, which demonstrate the expected second-order accuracy in the maximum norm over all grid points. Results now indicate that on these problems the IIM and XFEM methods give comparable accuracy in solution values. With appropriate interpolation it is also possible to obtain nearly second order accurate values of the solution and normal derivative at the interface with the IIM.
High-resolution rotated grid method for conservation laws with embedded geometries
(2005) Helzel, Christiane, 1971-; Berger, Marsha J.; LeVeque, Randall J
We develop a second-order rotated grid method for the approximation of time dependent solutions of conservation laws in complex geometry using an underlying Cartesian grid. Stability for time steps adequate for the regular part of the grid is obtained by increasing the domain of dependence of the numerical method near the embedded boundary by constructing h-boxes at grid cell interfaces. We describe a construction of h-boxes that not only guarantees stability but also leads to an accurate and conservative approximation of boundary cells that may be orders of magnitude smaller than regular grid cells. Of independent interest is the rotated difference scheme itself, on which the embedded boundary method is based.
A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions
(Society for Industrial and Applied Mathematics, 2002) LeVeque, Randall J; Bale, Derek S.; Mitran, Sorin; Rossmanith, James A.
We study a general approach to solving conservation laws of the form qt+f(q,x)x=0, where the flux function f(q,x) has explicit spatial variation. Finite-volume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A high-resolution wave-propagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)-f-1(Qi-1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be second-order accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws $q_t+f(q,x)_x=\psi(q,x)$ are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasi-steady problems close to steady state.
A class of approximate Riemann Solvers and their relation to relaxation schemes
(Elsevier/Academic Press, 2001-09-30) LeVeque, Randall J; Pelanti, Marica
We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math.48, 235 (1995)] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.
A wave propagation method for three-dimensional hyperbolic conservation laws
(Elsevier, 2000-11-20) Langseth, Jan Olav; LeVeque, Randall J
A class of wave propagation algorithms for three-dimensional conservation laws and other hyperbolic systems is developed. These unsplit finite-volume methods are based on solving one-dimensional Riemann problems at the cell interfaces and applying flux-limiter functions to suppress oscillations arising from second-derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse directions to model cross-derivative terms. With proper upwinding, a method that is stable for Courant numbers up to 1 can be developed. The stability theory for three-dimensional algorithms is found to be more subtle than in two dimensions and is studied in detail. In particular we find that some methods which are unconditionally unstable when no limiter is applied are (apparently) stabilized by the limiter function and produce good looking results. Several computations using the Euler equations are presented including blast wave and complex shock/vorticity problems. These algorithms are implemented in the software, which is freely available.
A wave propagation algorithm for hyperbolic systems on curved manifolds
(Elsevier, 2004-09-20) Rossmanith, James A.; Bale, Derek S.; LeVeque, Randall J
An extension of the wave propagation algorithm first introduced by LeVeque [J. Comput. Phys. 131 (1997) 327] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and high-resolution shock-capturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary one-dimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package and is freely available on the web.

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