Wave Equation In Second-order Form Research Articles

High-order accurate implicit-explicit time-stepping schemes for wave equations on overset grids

New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step, three levels in time, and based on the modified equation approach. Second and fourth-order accurate schemes are developed and they incorporate upwind dissipation for stability on overset grids. The fully implicit schemes are useful for certain applications such as the WaveHoltz algorithm for solving Helmholtz problems where very large time-steps are desired. Some wave propagation problems are geometrically stiff due to localized regions of small grid cells, such as grids needed to resolve fine geometric features, and for these situations the implicit time-stepping scheme is combined with an explicit scheme: the implicit scheme is used for component grids containing small cells while the explicit scheme is used on the other grids such as background Cartesian grids. The resulting partitioned implicit-explicit scheme can be many times faster than using an explicit scheme everywhere. The accuracy and stability of the schemes are studied through analysis and numerical computations.

Fractional-step finite difference schemes for incompressible elasticity on overset grids

Efficient finite-difference schemes for the numerical solution of the time-dependent equations of incompressible linear elasticity on complex geometry are presented. The schemes are second-order accurate in space and time, and solve the equations in displacement-pressure form. The algorithms use a fractional-step approach in which the time-step of the displacements is performed separately from the solution of a Poisson problem to update the pressure. Complex geometry is treated with curvilinear overset grids. Compatibility boundary conditions and an upwind dissipation for wave equations in second-order form are included to ensure stability for overset grids, and for the difficult case of traction (free-surface) boundary conditions. A divergence damping term is added to keep the dilatation small. The stability of the schemes is studied with GKS mode analysis. Exact eigen-mode solutions are obtained for several problems involving rectangular, cylindrical and spherical configurations, and these are valuable as benchmark problems. The accuracy and stability of the approach in two and three space dimensions is illustrated by comparing the numerical solutions with the exact solutions of the benchmark problems.

Modeling and inversion in acoustic-elastic coupled media using energy-stable summation-by-parts operators

Seismic data acquired at the seafloor are valuable in characterizing the subsurface and monitoring producing hydrocarbon fields. To fully use such data, a stable, accurate, and efficient numerical scheme is needed that accounts for acoustic and elastic wave propagation, their interaction at the seafloor interface, and for sources and receivers placed on either side of that interface. Existing methods either make incorrect assumptions or have high implementation and computational costs. We have developed a high-order finite-difference summation-by-parts framework for the acoustic-elastic wave equations in second-order form (in terms of displacements in the solid and an extension of velocity potential in the fluid). Our modified discretization of the elastic operator overcomes the dispersion errors known to plague displacement-based schemes in the high VP/ VS limit. We weakly impose boundary and interface conditions using simultaneous approximation terms, leading to an energy-stable numerical scheme that rigorously handles point injection and extraction. The fully discrete system is self-adjoint after a time reversal and a sign flip and is furthermore a high-order accurate discretization of the continuous problem. The self-adjointness ensures that forward and adjoint wavefields are computed with similar accuracy and simplifies gradient computation for inversion purposes. We find that our numerical scheme achieves accuracy comparable to a more computationally expensive spectral-element method and demonstrate its application to full-waveform inversion using the Marmousi2 model.

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations

We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.

Discontinuous Galerkin Galerkin Differences for the Wave Equation in Second-Order Form

We develop interior penalty discontinuous Galerkin difference methods for the wave equation in second-order form. The new schemes are energy conserving or energy dissipating depending on the simple...

Convergence analysis of an energy based discontinuous Galerkin method for the wave equation in second-order form: [formula omitted] version

In this paper we study the error estimates of the hp-version of a spatial discontinuous Galerkin method proposed in Appelö and Hagstrom (2015) for the wave equation. For different degrees of the approximation spaces we give the error estimates depending on the flux parameters in the energy norm and prove that optimal convergence rates can be obtained for some special flux parameters, which is an improvement of the existing work of Appelö and Hagstrom (2015). Finally we provide numerical experiments to verify our theoretical findings.

Dynamic rupture and earthquake sequence simulations using the wave equation in second-order form

SUMMARYWe present a numerical method for simulating both single-event dynamic ruptures and earthquake sequences with full inertial effects in antiplane shear with rate-and-state fault friction. We use the second-order form of the wave equation, expressed in terms of displacements, discretized with high-order-accurate finite difference operators in space. Advantages of this method over other methods include reduced computational memory usage and reduced spurious high frequency oscillations. Our method handles complex geometries, such as non-planar fault interfaces and free surface topography. Boundary conditions are imposed weakly using penalties. We prove time stability by constructing discrete energy estimates. We present numerical experiments demonstrating the stability and convergence of the method, and showcasing applications of the method, including the transition in rupture style from crack-like ruptures to slip pulses for strongly rate-weakening friction and the simulation of earthquake sequences in a viscoelastic solid with a fully dynamic coseismic phase.

Convergence Analysis of a Discontinuous Galerkin Method for Wave Equations in Second-Order Form

In this paper we study the convergence property of a spatial discontinuous Galerkin method for wave equations. We prove an optimal convergence rate in the energy norm, which improves an existing su...

High-order upwind schemes for the wave equation on overlapping grids: Maxwell's equations in second-order form

High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw [1] how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. The upwind scheme is shown to be robust and provide high-order accuracy.

A New Discontinuous Galerkin Formulation for Wave Equations in Second-Order Form

We develop and analyze a new strategy for the spatial discontinuous Galerkin discretization of wave equations in second-order form. The method features a direct, mesh-independent approach to defini...

Upwind schemes for the wave equation in second-order form

We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d’Alembert’s exact solution. We construct conservative finite difference approximations, although finite volume approximations are also possible. High-order accuracy is obtained using a space-time procedure which requires only two discrete time levels. The advantages of our approach include efficiency in both memory and speed together with accuracy and robustness. The stability and accuracy of the approximations in one and two space dimensions are studied through normal-mode analysis. The form of the dissipation and dispersion introduced by the schemes is elucidated from the modified equations. Upwind schemes are implemented and verified in one dimension for approximations up to sixth-order accuracy, and in two dimensions for approximations up to fourth-order accuracy. Numerical computations demonstrate the attractive properties of the approach for solutions with varying degrees of smoothness.

Spectral methods for the wave equation in second-order form

Current spectral simulations of Einstein's equations require writing the equations in first-order form, potentially introducing instabilities and inefficiencies. We present a new penalty method for pseudo-spectral evolutions of second order in space wave equations. The penalties are constructed as functions of Legendre polynomials and are added to the equations of motion everywhere, not only on the boundaries. Using energy methods, we prove semi-discrete stability of the new method for the scalar wave equation in flat space and show how it can be applied to the scalar wave on a curved background. Numerical results demonstrating stability and convergence for multi-domain second-order scalar wave evolutions are also presented. This work provides a foundation for treating Einstein's equations directly in second-order form by spectral methods.

Discrete boundary treatment for the shifted wave equation in second-order form and related problems

We present strongly stable semi-discrete finite difference approximations to the quarter space problem (x > 0, t > 0) for the first order in time, second order in space wave equation with a shift term. We consider space-like (pure outflow) and time-like boundaries, with either second- or fourth-order accuracy. These discrete boundary conditions suggest a general prescription for boundary conditions in finite difference codes approximating first order in time, second order in space hyperbolic problems, such as those that appear in numerical relativity. As an example we construct boundary conditions for the Nagy–Ortiz–Reula formulation of the Einstein equations coupled to a scalar field in spherical symmetry.

On coupled fields in stratified plasmas with tensor pressure perturbations — II

This paper deals with small-amplitude waves, described by Maxwell's equations and single-fluid hydrodynamics, in horizontally stratified, continuously varying plasmas with anisotropic electron pressure perturbations. First, two pairs of coupled wave equations in second-order form are obtained. Then, sets of equations in first-order matrix form are derived. One such set, which describes two kinds of vertically propagating transverse waves, is treated by the method of Clemmow and Heading. It is then transformed by Heading's method into a form which is valid throughout reflection regions, and from which a pair of second-order coupled wave equations in Forsterling's form is obtained.