Discretizing singular point sources in hyperbolic wave propagation problems

Petersson, N. Anders; OReilly, O. J.; Sjögreen, Björn; Bydlon, S. A.

doi:10.1016/j.jcp.2016.05.060

Cited by 31 publications

(28 citation statements)

References 28 publications

(69 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The interface conditions are of the same form as those considered in (98) and we may use the interface discretization presented in Theorem 3. References [16,21] describe how to discretize δ and ∂ x i δ. When using qth order finite difference stencils in the interior, the discretizations of δ and ∂ x i δ satisfy q and q + 1 moment conditions, respectively.…”

Section: Marine Seismic Exploration With Ocean-bottom Nodesmentioning

confidence: 99%

Non-stiff boundary and interface penalties for narrow-stencil finite difference approximations of the Laplacian on curvilinear multiblock grids

Almquist

Dunham

2020

Journal of Computational Physics

View full text Add to dashboard Cite

The Laplacian appears in several partial differential equations used to model wave propagation. Summation-by-parts-simultaneous approximation term (SBP-SAT) finite difference methods are often used for such equations, as they combine computational efficiency with provable stability on curvilinear multiblock grids. However, the existing SBP-SAT discretization of the Laplacian quickly becomes prohibitively stiff as grid skewness increases. The stiffness stems from the SATs that impose inter-block couplings and Dirichlet boundary conditions. We resolve this issue by deriving stable SATs whose stiffness is almost insensitive to grid skewness. The new discretization thus allows for large time steps in explicit time integrators, even on very skewed grids. It also applies to the variable-coefficient generalization of the Laplacian. We demonstrate the efficacy and versatility of the new method by applying it to acoustic wave propagation problems inspired by marine seismic exploration and infrasound monitoring of volcanoes.

show abstract

Section: Marine Seismic Exploration With Ocean-bottom Nodesmentioning

confidence: 99%

Non-stiff boundary and interface penalties for narrow-stencil finite difference approximations of the Laplacian on curvilinear multiblock grids

Almquist

Dunham

2020

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…Spurious oscillation can arise when source terms are introduced in schemes discretized by central finite differences. The triggering of spurious oscillations can be avoided by using Cartesian staggered grids, or by smoothing out the source discretization by imposing smoothness conditions, see [24]. However, there is no guarantee that spurious oscillations are avoided in our curvilinear staggered scheme.…”

Section: Point Source On the Boundarymentioning

confidence: 99%

“…In the above, δ(r 1 − r * ) is the Dirac distribution, centered at the source location r * , and s(t) is a given source time function. For a detailed discussion on how to discretize the Dirac distribution for hyperbolic problems, see [24]. To place the source on the boundary, we treat (47) as a boundary condition and impose it using a SAT term, see [22] for details.…”

Section: Point Source On the Boundarymentioning

confidence: 99%

Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

OReilly

Petersson

2020

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry and positive definiteness of the contravariant metric tensor. To improve accuracy and decrease computational cost, we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization. Bounds are derived that yield a point-wise condition that can be evaluated to check for stability of the modified discretization. This condition shows that the interpolation operators should be constructed such that their norm is close to unity.

show abstract

“…The geometry is discretized using two elastic blocks (one on each side of the crack) with (n + 1) × (n + 1) grid points and a single fluid block of size (n + 1) × (m + 1) , where n = 12 × 2 conditions (15). The manufactured solution in the fluid and the crack geometry are p(x, t) = sin(kx) cos(ωt), u(x, y, t) = sin(kx) sin(ky) cos(ωt) + sin(kx) cos(ωt), (38) (39) respectively. We prescribe the motion of the interface using The manufactured solution in the solid is…”

Section: Manufactured Solutionsmentioning

confidence: 99%

“…The latter, when the earthquakes are much smaller than modeled wavelengths, can be treated as point moment tensor sources. Details on how to discretize the singular source terms with high-order accuracy can be found in [39]. Here, for simplicity, we excite waves by Downloaded 08/27/17 to 130.236.83.247.…”

Section: Excitation In the Solidmentioning

confidence: 99%

Simulation of Wave Propagation Along Fluid-Filled Cracks Using High-Order Summation-by-Parts Operators and Implicit-Explicit Time Stepping

OReilly¹,

Dunham²,

Nordström³

2017

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

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 semidiscrete 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 highorder accuracy; additional simulations demonstrate applicability of the method to studies of wave propagation in and around branching hydraulic fractures.

show abstract

Discretizing singular point sources in hyperbolic wave propagation problems

Cited by 31 publications

References 28 publications

Non-stiff boundary and interface penalties for narrow-stencil finite difference approximations of the Laplacian on curvilinear multiblock grids

Non-stiff boundary and interface penalties for narrow-stencil finite difference approximations of the Laplacian on curvilinear multiblock grids

Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

Simulation of Wave Propagation Along Fluid-Filled Cracks Using High-Order Summation-by-Parts Operators and Implicit-Explicit Time Stepping

Contact Info

Product

Resources

About