Randall J. LeVeque

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, 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.

Brisa N Davis, Randall J LeVeque

Adaptive mesh refinement (AMR) is often used when solving time-dependent partial differential equations using numerical methods. It enables time-varying regions of much higher resolution, which can be used to track discontinuities in the solution by selectively refining around those areas. The open source Clawpack software implements block-structured AMR to refine around propagating waves in the AMRClaw package. For problems where the solution must be computed over a large domain but is only of interest in a small area this approach often refines waves that will not impact the target area. We seek a method that enables the identification and refinement of only the waves that will influence the target area. Here we show that solving the time-dependent adjoint equation and using a suitable inner product allows for a more precise refinement of the relevant waves. We present the adjoint methodology in general, and give details on how this method has been implemented in AMRClaw. Examples for linear acoustics equations are presented, and a computational performance analysis is conducted. The adjoint method is compared to AMR methods already available in the AMRClaw software, and the resulting advantages and disadvantages are discussed. The code for the examples presented is archived on Github.

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.