# A stable finite difference method for the elastic wave equation on complex geometries with free surfaces

## Abstract

The isotropic elastic wave equation governs the propagation of seismic waves caused by earthquakes and other seismic events. It also governs the propagation of waves in solid material structures and devices, such as gas pipes, wave guides, railroad rails and disc brakes. In the vast majority of wave propagation problems arising in seismology and solid mechanics there are free surfaces. These free surfaces have, in general, complicated shapes and are rarely flat. Another feature, characterizing problems arising in these areas, is the strong heterogeneity of the media, in which the problems are posed. For example, on the characteristic length scales of seismological problems, the geological structures of the earth can be considered piecewise constant, leading to models where the values of the elastic properties are also piecewise constant. Large spatial contrasts are also found in solid mechanics devices composed of different materials welded together. The presence of curved free surfaces, together with the typical strong material heterogeneity, makes the design of stable, efficient and accurate numerical methods for the elastic wave equation challenging. Today, many different classes of numerical methods are used for the simulation of elastic waves. Early on, most of the methods were based on finite difference approximationsmore »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 944324

- Report Number(s):
- UCRL-JRNL-237329

TRN: US200902%%686

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Journal Article

- Journal Name:
- Communications in Computational Physics, vol. 5, no. 1, January 1, 2009, pp. 84-107

- Additional Journal Information:
- Journal Volume: 5; Journal Issue: 1

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS; APPROXIMATIONS; BOUNDARY CONDITIONS; BRAKES; CURVILINEAR COORDINATES; DESIGN; EARTHQUAKES; ELASTICITY; FINITE DIFFERENCE METHOD; SEISMIC EVENTS; SEISMIC WAVES; SEISMOLOGY; SIMULATION; STABILITY; WAVE EQUATIONS; WAVE PROPAGATION

### Citation Formats

```
Appelo, D, and Petersson, N A.
```*A stable finite difference method for the elastic wave equation on complex geometries with free surfaces*. United States: N. p., 2007.
Web.

```
Appelo, D, & Petersson, N A.
```*A stable finite difference method for the elastic wave equation on complex geometries with free surfaces*. United States.

```
Appelo, D, and Petersson, N A. 2007.
"A stable finite difference method for the elastic wave equation on complex geometries with free surfaces". United States. https://www.osti.gov/servlets/purl/944324.
```

```
@article{osti_944324,
```

title = {A stable finite difference method for the elastic wave equation on complex geometries with free surfaces},

author = {Appelo, D and Petersson, N A},

abstractNote = {The isotropic elastic wave equation governs the propagation of seismic waves caused by earthquakes and other seismic events. It also governs the propagation of waves in solid material structures and devices, such as gas pipes, wave guides, railroad rails and disc brakes. In the vast majority of wave propagation problems arising in seismology and solid mechanics there are free surfaces. These free surfaces have, in general, complicated shapes and are rarely flat. Another feature, characterizing problems arising in these areas, is the strong heterogeneity of the media, in which the problems are posed. For example, on the characteristic length scales of seismological problems, the geological structures of the earth can be considered piecewise constant, leading to models where the values of the elastic properties are also piecewise constant. Large spatial contrasts are also found in solid mechanics devices composed of different materials welded together. The presence of curved free surfaces, together with the typical strong material heterogeneity, makes the design of stable, efficient and accurate numerical methods for the elastic wave equation challenging. Today, many different classes of numerical methods are used for the simulation of elastic waves. Early on, most of the methods were based on finite difference approximations of space and time derivatives of the equations in second order differential form (displacement formulation), see for example [1, 2]. The main problem with these early discretizations were their inability to approximate free surface boundary conditions in a stable and fully explicit manner, see e.g. [10, 11, 18, 20]. The instabilities of these early methods were especially bad for problems with materials with high ratios between the P-wave (C{sub p}) and S-wave (C{sub s}) velocities. For rectangular domains, a stable and explicit discretization of the free surface boundary conditions is presented in the paper [17] by Nilsson et al. In summary, they introduce a discretization, that use boundary-modified difference operators for the mixed derivatives in the governing equations. Nilsson et al. show that the method is second order accurate for problems with smoothly varying material properties and stable under standard CFL constraints, for arbitrarily varying material properties. In this paper we generalize the results of Nilsson et al. to curvilinear coordinate systems, allowing for simulations on non-rectangular domains. Using summation by parts techniques, we show that there exists a corresponding stable discretization of the free surface boundary condition on curvilinear grids. We also prove that the discretization is stable and energy conserving both in semi-discrete and fully discrete form. As for the Cartesian method in, [17], the stability and conservation results holds for arbitrarily varying material properties. By numerical experiments it is established that the method is second order accurate.},

doi = {},

url = {https://www.osti.gov/biblio/944324},
journal = {Communications in Computational Physics, vol. 5, no. 1, January 1, 2009, pp. 84-107},

number = 1,

volume = 5,

place = {United States},

year = {2007},

month = {12}

}