2024-03-29T11:51:46Zhttp://digital.lib.washington.edu/dspace-oai/requestoai:digital.lib.washington.edu:1773/21192020-09-02T18:08:41Zcom_1773_1969com_1773_3774col_1773_1977
A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions
LeVeque, Randall J
Bale, Derek S.
Mitran, Sorin
Rossmanith, James A.
finite-volume methods
high-resolution methods
conservation laws
source terms
discontinuous flux functions
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.
2005-10-05T16:47:25Z
2005-10-05T16:47:25Z
2005-10-05T16:47:25Z
2002
Preprint
SIAM Journal on Scientific Computing Volume 24, Number 3, pp. 955-978
1095-7197
http://hdl.handle.net/1773/2119
en
Society for Industrial and Applied Mathematics
oai:digital.lib.washington.edu:1773/21162020-09-02T18:10:34Zcom_1773_1969com_1773_3774col_1773_1977
A wave propagation algorithm for hyperbolic systems on curved manifolds
Rossmanith, James A.
Bale, Derek S.
LeVeque, Randall J
Wave propagation algorithms
High-resolution methods
Hyperbolic systems
Curved manifolds
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.
2005-10-04T16:44:57Z
2005-10-04T16:44:57Z
2005-10-04T16:44:57Z
2004-09-20
Preprint
Journal of Computational Physics Volume 199, Issue 2, pp. 631-662
0021-9991
http://hdl.handle.net/1773/2116
en_US
Elsevier
oai:digital.lib.washington.edu:1773/21182020-09-02T18:09:49Zcom_1773_1969com_1773_3774col_1773_1977
A class of approximate Riemann Solvers and their relation to relaxation schemes
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.
2005-10-04T16:45:25Z
2005-10-04T16:45:25Z
2005-10-04T16:45:25Z
2001-09-30
Preprint
Journal of Computational Physics Volume 172 Issue 2, pp. 572-591
0021-9991
http://hdl.handle.net/1773/2118
en_US
Elsevier/Academic Press
oai:digital.lib.washington.edu:1773/21172020-09-02T18:10:09Zcom_1773_1969com_1773_3774col_1773_1977
A wave propagation method for three-dimensional hyperbolic conservation laws
Langseth, Jan Olav
LeVeque, Randall J
finite-volume methods
high resolution
wave propagation
three dimensions
Euler equations
software
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.
2005-10-04T16:45:13Z
2005-10-04T16:45:13Z
2005-10-04T16:45:13Z
2000-11-20
Preprint
Journal of Computational Physics Volume 165, Issue 1, pp. 126-166
0021-9991
http://hdl.handle.net/1773/2117
en_US
Elsevier
oai:digital.lib.washington.edu:1773/46362020-09-02T18:07:42Zcom_1773_1969com_1773_3774col_1773_1977
High-resolution rotated grid method for conservation laws with embedded
geometries
Helzel, Christiane, 1971-
Berger, Marsha J.
LeVeque, Randall J
finite volume methods
conservation laws
Cartesian grids
irregular geometries
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.
2009-05-20T21:36:25Z
2009-05-20T21:36:25Z
2009-05-20T21:36:25Z
2005
Article
SIAM J. SCI. COMPUT. , Vol. 26, No. 3, pp. 785–809
http://hdl.handle.net/1773/4636
en_US
Copyright @ 2005 Society for Industrial and Applied Mathematics
oai:digital.lib.washington.edu:1773/46372020-09-02T18:07:18Zcom_1773_1969com_1773_3774col_1773_1977
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.
Beale, J. T.
(John Thomas), 1947-
Chopp, David
L.
LeVeque, Randall J
Li, Zhilin,
1956-
immersed interface
method (IIM)
elliptic interface
problems
finite difference
methods
discontinuous
coefficients
singular source
term
convergence
order
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.
2009-05-20T21:36:28Z
2009-05-20T21:36:28Z
2009-05-20T21:36:28Z
2008
Article
COMM. APP. MATH.
AND COMP. SCI., Vol. 3, No. 1, 2008
http://pjm.math.berkeley.edu/camcos/2008/3-1/p05.xhtml
http://hdl.handle.net/1773/4637
en_US
Copyright @ 2008
Mathematical Sciences Publishers
oai:digital.lib.washington.edu:1773/46382020-09-02T18:06:38Zcom_1773_1969com_1773_3774col_1773_1977
High-resolution finite volume methods for dusty gas jets and plumes
Pelanti, Marcia
LeVeque, Randall J
finite volume methods
hihg-resolution methods
volcanic flows
dusty gas
plumes
jets
shocks
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.
2009-05-20T21:36:30Z
2009-05-20T21:36:30Z
2009-05-20T21:36:30Z
2006
Article
SIAM J. SCI. COMPUT. , Vol. 28, No. 4, pp. 1115-1360
http://hdl.handle.net/1773/4638
en_US
Copyright @ 2006 Society for Industrial and Applied Mathematics
oai:digital.lib.washington.edu:1773/46392020-09-02T18:06:03Zcom_1773_1969com_1773_3774col_1773_1977
Finite volume methods and adaptive refinement for global tsunami propagation and local inundation
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.
2009-05-20T21:36:33Z
2009-05-20T21:36:33Z
2009-05-20T21:36:33Z
2006
Article
Science of Tsunami Hazards, Vol. 24, No. 5, pp 319-328
http://hdl.handle.net/1773/4639
en_US
Copyright 2006
oai:digital.lib.washington.edu:1773/223662020-09-02T18:05:39Zcom_1773_1969com_1773_3774col_1773_1977
Probabilistic Tsunami Hazard Assessment (PTHA) for Crescent City, CA. Final Report for Phase I
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.
2013-03-26T16:57:12Z
2013-03-26T16:57:12Z
2013-03-26T16:57:12Z
2013-02-02
Technical Report
http://hdl.handle.net/1773/22366
en_US
University of Washington Department of Applied Mathmatics
oai:digital.lib.washington.edu:1773/361912020-09-02T18:21:50Zcom_1773_1969com_1773_3774col_1773_1977
Tsunami Hazard Assessment of the Strait of Juan de Fuca
Gonzalez, Frank I.
LeVeque, Randall J
Adams, Loyce M.
Cascadia subduction zone (CSZ), adaptive mesh refinement, GeoClaw
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.
2016-05-23T15:42:51Z
2016-05-23T15:42:51Z
2016-05-23T15:42:51Z
2015-09-24
Technical Report
http://hdl.handle.net/1773/36191
en_US
oai:digital.lib.washington.edu:1773/418862020-09-02T18:04:03Zcom_1773_1969com_1773_3774col_1773_1977
GeoClaw Model Tsunamis Compared to Tide Gauge Results Final Report
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.
2018-06-05T17:24:16Z
2018-06-05T17:24:16Z
2018-06-05T17:24:16Z
2018-11-03
Technical Report
http://hdl.handle.net/1773/41886
en_US
oai:digital.lib.washington.edu:1773/431812020-09-02T18:03:22Zcom_1773_1969com_1773_3774col_1773_1977
Developing a Warning System for Inbound Tsunamis from the Cascadia Subduction Zone
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.
tsunami, earthquake, subduction zone, early warning
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.
2019-01-07T21:56:29Z
2019-01-07T21:56:29Z
2019-01-07T21:56:29Z
2018-01-07
Article
http://hdl.handle.net/1773/43181
en_US
oai:digital.lib.washington.edu:1773/438272020-09-02T18:02:38Zcom_1773_1969com_1773_3774col_1773_1977
Issues Encountered with ASCE Compatibility Criteria
Adams, Loyce
Gonzalez, Frank
LeVeque, Randall J
tsunami hazard analysis
ASCE 7-16
vertical evacuation structures
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.
2019-08-09T22:35:14Z
2019-08-09T22:35:14Z
2019-08-09T22:35:14Z
2019-05
Technical Report
http://hdl.handle.net/1773/43827
en_US
http://creativecommons.org/licenses/by/3.0/us/
Attribution 3.0 United States
oai:digital.lib.washington.edu:1773/455862020-09-02T17:59:09Zcom_1773_1969com_1773_3774col_1773_1977
Tsunami Hazard Assessment of Whatcom County, Washington. Project Report - Version 2
Adams, Loyce
LeVeque, Randall J
Gonzalez, Frank
tsunami hazard analysis
GeoClaw
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.
2020-06-29T23:18:19Z
2020-06-29T23:18:19Z
2020-06-29T23:18:19Z
2019-05-19
Technical Report
http://hdl.handle.net/1773/45586
en_US
http://creativecommons.org/licenses/by/3.0/us/
Attribution 3.0 United States
oai:digital.lib.washington.edu:1773/455872020-09-02T18:21:30Zcom_1773_1969com_1773_3774col_1773_1977
Probabilistic Source Selection for the Cascadia Subduction Zone
Adams, Loyce
LeVeque, Randall J
Rim, Donsub
Gonzalez, Frank I
tsunami hazard analysis
GeoClaw
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.
2020-06-29T23:24:48Z
2020-06-29T23:24:48Z
2020-06-29T23:24:48Z
2017-03-19
Technical Report
http://hdl.handle.net/1773/45587
en_US
http://creativecommons.org/licenses/by/3.0/us/
Attribution 3.0 United States
oai:digital.lib.washington.edu:1773/462582020-10-01T10:25:04Zcom_1773_1969com_1773_3774col_1773_1977
Accelerating invasions and the asymptotics of fat-tailed dispersal
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.
2020-09-30T21:38:58Z
2020-09-30T21:38:58Z
2020-09-30T21:38:58Z
2019
Article
http://hdl.handle.net/1773/46258