Randall J. LeVeque

Marsha J. Berger, David L. George, Randall J. LeVeque et al.

Many geophysical flow or wave propagation problems can be modeled with two-dimensional depth-averaged equations, of which the shallow water equations are the simplest example. We describe the GeoClaw software that has been designed to solve problems of this nature, consisting of open source Fortran programs together with Python tools for the user interface and flow visualization. This software uses high-resolution shock-capturing finite volume methods on logically rectangular grids, including latitude--longitude grids on the sphere. Dry states are handled automatically to model inundation. The code incorporates adaptive mesh refinement to allow the efficient solution of large-scale geophysical problems. Examples are given illustrating its use for modeling tsunamis, dam break problems, and storm surge. Documentation and download information is available at www.clawpack.org/geoclaw

Brisa N. Davis, Randall J. LeVeque

One difficulty in developing numerical methods for tsunami modeling is the fact that solutions contain regions where much higher resolution is required than elsewhere in the domain, particularly since the solution may contain discontinuities or other localized features. The Clawpack software deals with this issue by using block-structured adaptive mesh refinement to selectively refine around propagating waves. For problems where only a target area of the total solution is of interest (e.g. one coastal community), a method that allows identifying and refining the grid only in regions that influence this target area would significantly reduce the computational cost of finding a solution. In this work, we show that solving the time-dependent adjoint equation and using a suitable inner product with the forward solution allows more precise refinement of the relevant waves. We present examples solving the shallow water equations in one and two dimensions. To perform these simulations, the use of the adjoint method has been integrated into the adaptive mesh refinement strategy of the open source GeoClaw software. We also present results that show that the accuracy of the solution is maintained and the computational time required is significantly reduced through the integration of the adjoint method into adaptive mesh refinement.

Brisa N. Davis, Randall J. LeVeque

One difficulty in developing numerical methods for hyperbolic systems of conservation laws is the fact that solutions often contain regions where much higher resolution is required than elsewhere in the domain, particularly since the solution may contain discontinuities or other localized features. The Clawpack software deals with this issue by using block-structured adaptive mesh refinement to selectively refine around propagating waves. For problems where only a target area of the total solution is of interest, a method that allows identifying and refining the grid only in regions that influence this target area would significantly reduce the computational cost of finding a solution. In this work, we show that solving the time-dependent adjoint equation and using a suitable inner product with the forward solution allows more precise refinement of the relevant waves. We present acoustics examples in one and two dimensions and a tsunami propagation example. To perform these simulations, the use of the adjoint method has been integrated into the adaptive mesh refinement strategy of the open source Clawpack and GeoClaw software. We also present results that show that the accuracy of the solution is maintained and the computational time required is significantly reduced through the integration of the adjoint method into AMR.

Donsub Rim, Scott Moe, Randall J. LeVeque

Snapshot matrices built from solutions to hyperbolic partial differential equations exhibit slow decay in singular values, whereas fast decay is crucial for the success of projection- based model reduction methods. To overcome this problem, we build on previous work in symmetry reduction [Rowley and Marsden, Physica D (2000), pp. 1-19] and propose an iterative algorithm that decomposes the snapshot matrix into multiple shifting profiles, each with a corresponding speed. Its applicability to typical hyperbolic problems is demonstrated through numerical examples, and other natural extensions that modify the shift operator are considered. Finally, we give a geometric interpretation of the algorithm.

Randall J. LeVeque, Knut Waagan, Frank I. González et al.

In order to perform probabilistic tsunami hazard assessment (PTHA) based on subduction zone earthquakes, it is necessary to start with a catalog of possible future events along with the annual probability of occurance, or a probability distribution of such events that can be easily sampled. For nearfield events, the distribution of slip on the fault can have a significant effect on the resulting tsunami. We present an approach to defining a probability distribution based on subdividing the fault geometry into many subfaults and prescribing a desired covariance matrix relating slip on one subfault to slip on any other subfault. The eigenvalues and eigenvectors of this matrix are then used to define a Karhunen-Loève expansion for random slip patterns. This is similar to a spectral representation of random slip based on Fourier series but conforms to a general fault geometry. We show that only a few terms in this series are needed to represent the features of the slip distribution that are most important in tsunami generation, first with a simple one-dimensional example where slip varies only in the down-dip direction and then on a portion of the Cascadia Subduction Zone.

David I. Ketcheson, Matteo Parsani, Randall J. LeVeque

We present a finite volume method that is applicable to hyperbolic PDEs including spatially varying and semilinear nonconservative systems. The spatial discretization, like that of the well-known Clawpack software, is based on solving Riemann problems and calculating fluctuations (not fluxes). The implementation employs weighted essentially non-oscillatory reconstruction in space and strong stability preserving Runge-Kutta integration in time. The method can be extended to arbitrarily high order of accuracy and allows a well-balanced implementation for capturing solutions of balance laws near steady state. This well-balancing is achieved through the $f$-wave Riemann solver and a novel wave-slope WENO reconstruction procedure. The wide applicability and advantageous properties of the method are demonstrated through numerical examples, including problems in nonconservative form, problems with spatially varying fluxes, and problems involving near-equilibrium solutions of balance laws.

Mauricio J. Del Razo, Yoichi Morofuji, James S. Meabon et al.

There is growing concern that blast-exposed individuals are at risk of developing neurological disorders later in life. Therefore, it is important to understand the dynamic properties of blast forces on brain cells, including the endothelial cells that maintain the blood-brain barrier (BBB), which regulates the passage of nutrients into the brain and protects it from toxins in the blood. To better understand the effect of shock waves on the BBB we have investigated an {\em in vitro} model in which BBB endothelial cells are grown in transwell vessels and exposed in a shock tube, confirming that BBB integrity is directly related to shock wave intensity. It is difficult to directly measure the forces acting on these cells in the transwell container during the experiments, and so a computational tool has been developed and presented in this paper. Two-dimensional axisymmetric Euler equations with the Tammann equation of state were used to model the transwell materials, and a high-resolution finite volume method based on Riemann solvers and the Clawpack software was used to solve these equations in a mixed Eulerian/Lagrangian frame. Results indicated that the geometry of the transwell plays a significant role in the observed pressure time series in these experiments. We also found that pressures can fall below vapor pressure due to the interaction of reflecting and diffracting shock waves, suggesting that cavitation bubbles could be a damage mechanism. Computations that include a simulated hydrophone inserted in the transwell suggest that the instrument itself could significantly alter blast wave properties. These findings illustrate the need for further computational modeling studies aimed at understanding possible blast-induced BBB damage.

David I Ketcheson, Randall J. LeVeque

Solutions of constant-coefficient nonlinear hyperbolic PDEs generically develop shocks, even if the initial data is smooth. Solutions of hyperbolic PDEs with variable coefficients can behave very differently. We investigate formation and stability of shock waves in a one-dimensional periodic layered medium by computational study of time-reversibility and entropy evolution. We find that periodic layered media tend to inhibit shock formation. For small initial conditions and large impedance variation, no shock formation is detected even after times much greater than the time of shock formation in a homogeneous medium. Furthermore, weak shocks are observed to be dynamically unstable in the sense that they do not lead to significant long-term entropy decay. We propose a characteristic condition for admissibility of shocks in heterogeneous media that generalizes the classical Lax entropy condition and accurately predicts the formation or absence of shocks in these media.

Kirsten Fagnan, Randall J. LeVeque, Thomas J. Matula

Extracorporeal Shock Wave Therapy (ESWT) is a noninvasive treatment for a variety of musculoskeletal ailments. A shock wave is generated in water and then focused using an acoustic lens or reflector so the energy of the wave is concentrated in a small treatment region where mechanical stimulation enhances healing. In this work we have computationally investigated shock wave propagation in ESWT by solving a Lagrangian form of the isentropic Euler equations in the fluid and linear elasticity in the bone using high-resolution finite volume methods. We solve a full three-dimensional system of equations and use adaptive mesh refinement to concentrate grid cells near the propagating shock. We can model complex bone geometries, the reflection and mode conversion at interfaces, and the the propagation of the resulting shear stresses generated within the bone. We discuss the validity of our simplified model and present results validating this approach.

Mauricio J. Del Razo, Randall J. LeVeque

Many experiments in biomedical applications and other disciplines use a shock tube. These experiments often involve placing an experimental sample within a fluid-filled container, which is then placed inside the shock tube. The shock tube produces an initial shock that propagates through gas before hitting the container with the sample. In order to gain insight into the shock dynamics that is hard to obtain by experimental means, computational simulations of the shock wave passing from gas into a thin elastic solid and into a nearly incompressible fluid are developed. It is shown that if the solid interface is very thin, it can be neglected, simplifying the model. The model uses Euler equations for compressible fluids coupled with a Tammann equation of state (EOS) to model both compressible gas and almost incompressible materials. A three-dimensional (2D axisymmetric) model of these equations is solved using high-resolution shock-capturing methods, with newly developed Riemann solvers and limiters. The methods are extended to work on a mapped grid to allow more complicated interface geometry, and they are adapted to work with adaptive mesh refinement (AMR) for higher resolution and faster computations. The Clawpack software is used to implement the method. These methods were initially inspired by shock tube experiments to study the injury mechanisms of traumatic brain injury (TBI).

Mauricio J. Del Razo, Randall J. LeVeque

The following study is motivated by experimental studies in traumatic brain injury (TBI). Recent research has demonstrated that low intensity non-impact blast wave exposure frequently leads to mild traumatic brain injury (mTBI); however, the mechanisms connecting the blast waves and the mTBI remain unclear. Collaborators at the Seattle VA Hospital are doing experiments to understand how blast waves can produce mTBI. In order to gain insight that is hard to obtain by experimental means, we have developed conservative finite volume methods for interface-shock wave interaction to simulate these experiments. A 1D model of their experimental setup has been implemented using Euler equations for compressible fluids. These equations are coupled with a Tammann equation of state (EOS) that allows us to model compressible gas along with almost incompressible fluids or elastic solids. A hybrid HLLC-exact Eulerian-Lagrangian Riemann solver for Tammann EOS with a jump in the parameters has been developed. The model has shown that if the plastic interface is very thin, it can be neglected. This result might be very helpful to model more complicated setups in higher dimensions.

Reika Nomura, Louise A. Hirao Vermare, Saneiki Fujita et al.

Although the sequential tsunami scenario detection framework was validated in our previous work, several tasks remain to be resolved from a practical point of view. This study aims to evaluate the performance of the previous tsunami scenario detection framework using a diverse database consisting of complex fault rupture patterns with heterogeneous slip distributions. Specifically, we compare the effectiveness of scenario superposition to that of the previous most likely scenario detection method. Additionally, how the length of the observation time window influences the accuracy of both methods is analyzed. We utilize an existing database comprising 1771 tsunami scenarios targeting the city Westport (WA, U.S.), which includes synthetic wave height records and inundation distributions as the result of fault rupture in the Cascadia subduction zone. The heterogeneous patterns of slips used in the database increase the diversity of the scenarios and thus make it a proper database for evaluating the performance of scenario superposition. To assess the performance, we consider various observation time windows shorter than 15 minutes and divide the database into five testing and learning sets. The evaluation accuracy of the maximum offshore wave, inundation depth, and its distribution is analyzed to examine the advantages of the scenario superposition method over the previous method. We introduce the dynamic time warping (DTW) method as an additional benchmark and compare its results to that of the Bayesian scenario detection method.

M. E. M. Arcos, Randall J. LeVeque

The ability to measure, predict, and compute tsunami flow velocities is of importance in risk assessment and hazard mitigation. Substantial damage can be done by high velocity flows, particularly in harbors and bays, even when the wave height is small. Moreover, advancing the study of sediment transport and tsunami deposits depends on the accurate interpretation and modeling of tsunami flow velocities and accelerations. Until recently, few direct measurements of tsunami velocities existed to compare with model results. During the 11 March 2011 Tohoku Tsunami 328 current meters were in place around the Hawaiian Islands, USA, that captured time series of water velocity in 18 locations, in both harbors and deep channels, at a series of depths. We compare several of these velocity records against numerical simulations performed using the GeoClaw numerical tsunami model, based on solving the depth-averaged shallow water equations with adaptive mesh refinement, to confirm that this model can accurately predict velocities at nearshore locations. Model results demonstrate tsunami current velocity is more spatially variable than wave form or height and therefore may be a more sensitive variable for model validation.