N. Anders Petersson

Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation

We consider the three-dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress-free boundary condition. Building on our recently developed theory for difference methods for second order hyperbolic systems [H.-O. Kreiss, N. A. Petersson, J. Ystrom, SIAM J. Numer. Anal., 40 (2002), pp. 1940-1967], we develop an explicit, second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities. The spatial discretization is self-adjoint, and the stability is obtained through an energy estimate. Seismic events are often modeled using singular source terms, and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source. Several numerical examples are given.

A Second Order Accurate Embedded Boundary Method for the Wave Equation with Dirichlet Data

The accuracy of Cartesian embedded boundary methods for the second order wave equation in general two-dimensional domains subject to Dirichlet boundary conditions is analyzed. Based on the analysis, we develop a numerical method where both the solution and its gradient are second order accurate. We avoid the small-cell stiffness problem without sacrificing the second order accuracy by adding a small artificial term to the Dirichlet boundary condition. Long-time stability of the method is obtained by adding a small fourth order dissipative term. Several numerical examples are provided to demonstrate the accuracy and stability of the method. The method is also used to solve the two-dimensional TM

Difference Approximations of the Neumann Problem for the Second Order Wave Equation

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A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation

problem for Maxwells equations posed as a second order wave equation for the electric field coupled to ordinary differential equations for the magnetic field.

An Embedded Boundary Method for the Wave Equation with Discontinuous Coefficients

Stability theory and numerical experiments are presented for a finite difference method that directly discretizes the Neumann problem for the second order wave equation. Complex geometries are discretized using a Cartesian embedded boundary technique. Both second and third order accurate approximations of the boundary conditions are presented. Away from the boundary, the basic second order method can be corrected to achieve fourth order spatial accuracy. To integrate in time, we present both a second order and a fourth order accurate explicit method. The stability of the method is ensured by adding a small fourth order dissipation operator, locally modified near the boundary to allow its application at all grid points inside the computational domain. Numerical experiments demonstrate the accuracy and long-time stability of the proposed method.

Boundary estimates for the elastic wave equation in almost incompressible materials

We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (μ(x)ux)x. This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (μ(x)ux)x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.

Discretizing singular point sources in hyperbolic wave propagation problems

A second order accurate embedded boundary method for the two-dimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is allowed to intersect the mesh in an arbitrary fashion. By using ghost points on either side of the interface, previous embedded boundary techniques for the Neumann and Dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy. The resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points, resulting in a fully explicit time-integration method. We prove that the one-dimensional restriction of the method is stable without damping for arbitrary locations of the interface relative to the grid. For the two-dimensional case, the previously developed fourth order

Wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation-by-parts finite difference method

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Source Estimation by Full Wave Form Inversion

-dissipation is generalized to handle jump conditions. We demonstrate that this operator provides sufficient stabilization to enable long-time simulations while being weak enough to preserve the accuracy of the solution. Numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder. The numerical solutions are evaluated against the analytical solution due to Mie, and pointwise second order accuracy is confirmed.

Toward Exascale Earthquake Ground Motion Simulations for Near-Fault Engineering Analysis

We study the half-plane problem for the elastic wave equation subject to a free surface boundary condition, with particular emphasis on almost incompressible materials. A normal mode analysis is developed to estimate the solution in terms of the boundary data, showing that the problem is boundary stable. The dependence on the material properties, which is difficult to analyze by the energy method, is made transparent by our estimates. The normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation. Our analysis explains why the number of grid points per wave length must be increased when the shear modulus (