Fast, accurate solutions for curvilinear earthquake faults and anelastic strain

Imaging the anelastic deformation within the crust and lithosphere using surface geophysical data remains a significant challenge in part due to the wide range of physical processes operating at different depths and to various levels of localization that they embody. Models of Earth's elastic properties from seismological imaging combined with geodetic modeling may form the basis of comprehensive rheological models of Earth's interior. However, representing the structural complexity of faults and shear zones in numerical models of deformation still constitutes a major difficulty. Here, we present numerical techniques for high-precision models of deformation and stress around both curvilinear faults and volumes undergoing anelastic (irreversible) strain in a heterogenous elastic half-space. To that end, we enhance the software Gamra to model triangular and rectangular fault patches and tetrahedral and cuboidal strain volumes. This affords a means of rapid and accurate calculations of elasto-static Green's functions for localized (e.g., faulting) and distributed (e.g., viscoelastic) deformation in Earth's crust and lithosphere. We demonstrate the correctness of the method with analytic tests, and we illustrate its practical performance by solving for coseismic and postseismic deformation following the 2015 Mw 7.8 Gorkha, Nepal earthquake to extremely high precision

Gamra: Simple meshing for complex earthquakes

The static offsets caused by earthquakes are well described by elastostatic models with a discontinuity in the displacement along the fault. A traditional approach to model this discontinuity is to align the numerical mesh with the fault and solve the equations using finite elements. However, this distorted mesh can be difficult to generate and update. We present a new numerical method, inspired by the Immersed Interface Method (Leveque and Li, 1994), for solving the elastostatic equations with embedded discontinuities. This method has been carefully designed so that it can be used on parallel machines on an adapted finite difference grid. We have implemented this method in Gamra, a new code for earth modeling. We demonstrate the correctness of the method with analytic tests, and we demonstrate its practical performance by solving a realistic earthquake model to extremely high precision

Scaling the semidefinite program solver SDPB

We present enhancements to SDPB, an open source, parallelized, arbitrary precision semidefinite program solver designed for the conformal bootstrap. The main enhancement is significantly improved performance and scalability using the Elemental library and MPI. The result is a new version of SDPB that runs on multiple nodes with hundreds of cores with excellent scaling, making it practical to solve larger problems. We demonstrate performance on a moderate-size problem in the 3d Ising CFT and a much larger problem in the $O(2)$ Model.Comment: 13 pages plus references, 2 figure

Growth of perturbations in homogeneous collisionless collapse: discs versus spheres

We study the growth of radial perturbations during the collapse of homogeneous spheres and discs. We model the sphere by concentric shells and the disc by concentric rings, initially at rest and allowed to move only radially. Our numerical experiments show only moderate growth in spheres but rapid growth in discs, leading to pronounced ring formation. For a sphere, the perturbations can be treated analytically as in cosmology. The growth rate is that of a power-law in time and independent of scale. For discs, we present an analytic solution for the perturbations using a suitable mode expansion. Shorter wavelength perturbations grow faster, exhibiting exponential growth at early times

Fast, accurate solutions for curvilinear earthquake faults and anelastic strain

Imaging the anelastic deformation within the crust and lithosphere using surface geophysical data remains a significant challenge in part due to the wide range of physical processes operating at different depths and to various levels of localization that they embody. Models of Earth's elastic properties from seismological imaging combined with geodetic modeling may form the basis of comprehensive rheological models of Earth's interior. However, representing the structural complexity of faults and shear zones in numerical models of deformation still constitutes a major difficulty. Here, we present numerical techniques for high-precision models of deformation and stress around both curvilinear faults and volumes undergoing anelastic (irreversible) strain in a heterogenous elastic half-space. To that end, we enhance the software Gamra to model triangular and rectangular fault patches and tetrahedral and cuboidal strain volumes. This affords a means of rapid and accurate calculations of elasto-static Green's functions for localized (e.g., faulting) and distributed (e.g., viscoelastic) deformation in Earth's crust and lithosphere. We demonstrate the correctness of the method with analytic tests, and we illustrate its practical performance by solving for coseismic and postseismic deformation following the 2015 Mw 7.8 Gorkha, Nepal earthquake to extremely high precision

Development, verification, and maintenance of computational software in geodynamics

Research on dynamical processes within the Earth and planets increasingly relies upon sophisticated, large-scale computational models. Improved understanding of fundamental physical processes such as mantle convection and the geodynamo, magma dynamics, crustal and lithospheric deformation, earthquake nucleation, and seismic wave propagation, are heavily dependent upon better numerical modeling. Surprisingly, the rate-limiting factor for progress in these areas is not just computing hardware, as was once the case. Rather, advances in software are not keeping pace with the recent improvements in hardware. Modeling tools in geophysics are usually developed and maintained by individual scientists, or by small groups. But it is difficult for any individual, or even a small group, to keep up with sweeping advances in computing hardware, parallel processing software, and numerical modeling methodology