| 1. |
- Erickson, Brittany A., et al.
(författare)
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Accuracy of Stable, High-order Finite Difference Methods for Hyperbolic Systems with Non-smooth Wave Speeds
- 2019
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Ingår i: Journal of Scientific Computing. - : Springer-Verlag New York. - 0885-7474 .- 1573-7691. ; 81:3, s. 2356-2387
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Tidskriftsartikel (refereegranskat)abstract
- We derive analytic solutions to the scalar and vector advection equation with variable coefficients in one spatial dimension using Laplace transform methods. These solutions are used to investigate how accuracy and stability are influenced by the presence of discontinuous wave speeds when applying high-order-accurate, skew-symmetric finite difference methods designed for smooth wave speeds. The methods satisfy a summation-by-parts rule with weak enforcement of boundary conditions and formal order of accuracy equal to 2, 3, 4 and 5. We study accuracy, stability and convergence rates for linear wave speeds that are (a) constant, (b) non-constant but smooth, (c) continuous with a discontinuous derivative, and (d) constant with a jump discontinuity. Cases (a) and (b) correspond to smooth wave speeds and yield stable schemes and theoretical convergence rates. Non-smooth wave speeds [cases (c) and (d)], however, reveal reductions in theoretical convergence rates and in the latter case, the presence of an instability.
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| 2. |
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| 3. |
- O’Reilly, Ossian, et al.
(författare)
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Energy stable and high-order-accurate finite difference methods on staggered grids
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- For wave propagation over distances of many wavelengths, high-order finite difference methods on staggered grids are widely used due to their excellent dispersion properties. However, the enforcement of boundary conditions in a stable manner and treatment of interface problems with discontinuous coefficients usually pose many challenges. In this work, we construct a provably stable and high-order-accurate finite difference method on staggered grids that can be applied to a broad class of boundary and interface problems. The staggered grid difference operators are in summation-by-parts form and when combined with a weak enforcement of the boundary conditions, lead to an energy stable method on multiblock grids. The general applicability of the method is demonstrated by simulating an explosive acoustic source, generating waves reflecting against a free surface and material discontinuity.
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| 4. |
- O'Reilly, Ossian, et al.
(författare)
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Energy stable and high-order-accurate finite difference methods on staggered grids
- 2017
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Ingår i: Journal of Computational Physics. - : Academic Press. - 0021-9991 .- 1090-2716. ; 346, s. 572-589
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Tidskriftsartikel (refereegranskat)abstract
- For wave propagation over distances of many wavelengths, high-order finite difference methods on staggered grids are widely used due to their excellent dispersion properties. However, the enforcement of boundary conditions in a stable manner and treatment of interface problems with discontinuous coefficients usually pose many challenges. In this work, we construct a provably stable and high-order-accurate finite difference method on staggered grids that can be applied to a broad class of boundary and interface problems. The staggered grid difference operators are in summation-by-parts form and when combined with a weak enforcement of the boundary conditions, lead to an energy stable method on multiblock grids. The general applicability of the method is demonstrated by simulating an explosive acoustic source, generating waves reflecting against a free surface and material discontinuity.
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| 5. |
- O'Reilly, Ossian, 1986-
(författare)
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Numerical methods for wave propagation in solids containing faults and fluid-filled fractures
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis develops numerical methods for the simulation of wave propagation in solids containing faults and fluid-filled fractures. These techniques have applications in earthquake hazard analysis, seismic imaging of reservoirs, and volcano seismology. A central component of this work is the coupling of mechanical systems. This aspect involves the coupling of both ordinary differential equations (ODE)(s) and partial differential equations (PDE)(s) along curved interfaces. All of these problems satisfy a mechanical energy balance. This mechanical energy balance is mimicked by the numerical scheme using high-order accurate difference approximations that satisfy the principle of summation by parts, and by weakly enforcing the coupling conditions. The first part of the thesis considers the simulation of dynamic earthquake ruptures along non-planar fault geometries and the simulation of seismic wave radiation from earthquakes, when the earthquakes are idealized as point moment tensor sources. The dynamic earthquake rupture process is simulated by coupling the elastic wave equation at a fault interface to nonlinear ODEs that describe the fault mechanics. The fault geometry is complex and treated by combining structured and unstructured grid techniques. In other applications, when the earthquake source dimension is smaller than wavelengths of interest, the earthquake can be accurately described by a point moment tensor source localized at a single point. The numerical challenge is to discretize the point source with high-order accuracy and without producing spurious oscillations.The second part of the thesis presents a numerical method for wave propagation in and around fluid-filled fractures. This problem requires the coupling of the elastic wave equation to a fluid inside curved and branching fractures in the solid. The fluid model is a lubrication approximation that incorporates fluid inertia, compressibility, and viscosity. The fracture geometry can have local irregularities such as constrictions and tapered tips. The numerical method discretizes the fracture geometry by using curvilinear multiblock grids and applies implicit-explicit time stepping to isolate and overcome stiffness arising in the semi-discrete equations from viscous diffusion terms, fluid compressibility, and the particular enforcement of the fluid-solid coupling conditions. This numerical method is applied to study the interaction of waves in a fracture-conduit system. A methodology to constrain fracture geometry for oil and gas (hydraulic fracturing) and volcano seismology applications is proposed.The third part of the thesis extends the summation-by-parts methodology to staggered grids. This extension reduces numerical dispersion and enables the formulation of stable and high-order accurate multiblock discretizations for wave equations in first order form on staggered grids. Finally, the summation-by-parts methodology on staggered grids is further extended to second derivatives and used for the treatment of coordinate singularities in axisymmetric wave propagation.
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| 6. |
- O'Reilly, Ossian, et al.
(författare)
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Provably non-stiff implementation of weak coupling conditions for hyperbolic problems
- 2022
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Ingår i: Numerische Mathematik. - Heidelberg : Springer. - 0029-599X .- 0945-3245. ; :150, s. 551-589
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Tidskriftsartikel (refereegranskat)abstract
- In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant-Friedrichs-Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals.
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| 7. |
- O’Reilly, Ossian, et al.
(författare)
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Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods
- 2015
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Ingår i: Communications in Computational Physics. - : Global Science Press. - 1815-2406 .- 1991-7120. ; 17:2, s. 337-370
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Tidskriftsartikel (refereegranskat)abstract
- We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite difference methods are in summation-by-parts form. Interface conditions coupling the numerical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneousapproximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the particle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.
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| 8. |
- O'Reilly, Ossian, et al.
(författare)
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Simulation of Wave Propagation Along Fluid-Filled Cracks Using High-Order Summation-by-Parts Operators and Implicit-Explicit Time Stepping
- 2017
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM PUBLICATIONS. - 1064-8275 .- 1095-7197. ; 39:4, s. B675-B702
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Tidskriftsartikel (refereegranskat)abstract
- We present an ecient, implicit-explicit numerical method for wave propagation insolids containing uid-lled cracks, motivated by applications in geophysical imaging of fracturedoil/gas reservoirs and aquifers, volcanology, and mechanical engineering. We couple the elastic waveequation in the solid to an approximation of the linearized, compressible Navier{Stokes equationsin curved and possibly branching cracks. The approximate uid model, similar to the widely usedlubrication model but accounting for uid inertia and compressibility, exploits the narrowness of thecrack relative to wavelengths of interest. The governing equations are spatially discretized usinghigh-order summation-by-parts nite dierence operators and the uid-solid coupling conditions areweakly enforced, leading to a provably stable scheme. Stiness of the semidiscrete equations can arisefrom the enforcement of coupling conditions, uid compressibility, and diusion operators requiredto capture viscous boundary layers near the crack walls. An implicit-explicit Runge{Kutta scheme isused for time stepping, and the entire system of equations can be advanced in time with high-orderaccuracy using the maximum stable time step determined solely by the standard CFL restriction forwave propagation, irrespective of the crack geometry and uid viscosity. The uid approximationleads to a sparse block structure for the implicit system, such that the additional computationalcost of the uid is small relative to the explicit elastic update. Convergence tests verify highorderaccuracy; additional simulations demonstrate applicability of the method to studies of wavepropagation in and around branching hydraulic fractures.
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| 9. |
- Oreilly, Ossian, et al.
(författare)
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Simultation of wave propagation along fluid-filled cracks using high-order summation-by-parts operators and implicit-explicit time stepping
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We present an efficient, implicit-explicit numerical method for wave propagation in solids containing fluid-filled cracks, motivated by applications in geophysical imaging of fractured oil/gas reservoirs and aquifers, volcanology, and mechanical engineering. We couple the elastic wave equation in the solid to an approximation of the linearized, compressible Navier-Stokes equations in curved and possibly branching cracks. The approximate fluid model, similar to the widely used lubrication model but accounting for fluid inertia and compressibility, exploits the narrowness of the crack relative to wavelengths of interest. The governing equations are spatially discretized using high-order summation-by-parts finite difference operators and the fluid-solid coupling conditions are weakly enforced, leading to a provably stable scheme.Stiffness of the semi-discrete equations can arise from the enforcement of coupling conditions, fluid compressibility, and diffusion operators required to capture viscous boundary layers near the crack walls. An implicit-explicit Runge-Kutta scheme is used for time stepping and the entire system of equations can be advanced in time with high-order accuracy using the maximum stable time step determined solely by the standard CFL restriction for wave propagation, irrespective of the crack geometry and fluid viscosity. The fluid approximation leads to a sparse block structure for the implicit system, such that the additional computational cost of the fluid is small relative to the explicit elastic update. Convergence tests verify high-order accuracy; additional simulations demonstrate applicability of the method to studies of wave propagation in and around branching hydraulic fractures.
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| 10. |
- Petersson, N. Anders, et al.
(författare)
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Discretizing singular point sources in hyperbolic wave propagation problems
- 2016
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 321, s. 532-555
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Tidskriftsartikel (refereegranskat)abstract
- We develop high order accurate source discretizations for hyperbolic wave propagation problems in first order formulation that are discretized by finite difference schemes. By studying the Fourier series expansions of the source discretization and the finite difference operator, we derive sufficient conditions for achieving design accuracy in the numerical solution. Only half of the conditions in Fourier space can be satisfied through moment conditions on the source discretization, and we develop smoothness conditions for satisfying the remaining accuracy conditions. The resulting source discretization has compact support in physical space, and is spread over as many grid points as the number of moment and smoothness conditions. In numerical experiments we demonstrate high order of accuracy in the numerical solution of the 1-D advection equation (both in the interior and near a boundary), the 3-D elastic wave equation, and the 3-D linearized Euler equations.
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