## Wave Propagation

Waves appear in many different forms in nature, such as in acoustics, seismology, electromagnetism, oceanography, and general relativity. In this project, we develop accurate methods for wave propagation problems.

### Participants

- Gunilla Kreiss
- Gustav Ludvigsson
- Jonatan Werpers
- Ken Mattsson
- Martin Almquist
- Simon Sticko
- Siyang Wang
- Ylva Rydin

### Models

#### The Acoustic wave equation

The propagation of sound waves in the atmosphere is accurately described by the acoustic wave equation. Thus, one can use the wave equation to predict how noise from wind turbines, roads, airports, etc. spreads. However, since refraction in the atmosphere and reflection at the ground influence the propagation greatly, complicated 3-D geometries must be handled.

#### The Dirac equation

In many evolving applications in materials science involving for instance graphene, thermoelectric materials or heavy elements in catalytic materials science, the Schrödinger equation does not capture the fundamental physics of the electrons in the material. Here one instead needs the fundamental equation in relativistic quantum mechanics: the Dirac equation.

One of the main interests in this project is to improve the stability and accuracy for relativistic quantum mechanical calculations applied to problems relevant to materials science. Thus far, a strictly stable high-order accurate finite difference method for the Dirac equation has been constructed and implemented. We will now focus on applying the Dirac-solver to Klein-tunneling phenomena in graphene and the outer electron behaviour in heavy elements relevant to catalytic materials science. In particular, time-dependent phenomena such as Klein tunneling, Aharonov-Bohm effects, and local polarizations such as Berry phases will be investigated.

#### The Elastic wave equation

Distinct features of elastic wave propagation appear at boundaries and interfaces. Examples include the Rayleigh surface wave at a traction free surface and the Stoneley interface wave at an interface between two elastic materials in contact. Boundary phenomena are often involve sudden changes in present phase velocities and spatial frequencies. This project focuses on high order accurate discretizations of boundary and interface conditions for the elastic wave equation.

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The Stoneley interface wave at the interface between two elastic materials |

### Methods

#### Finite Differences

High-order finite difference methods are often very efficient for wave propagation problems. To ensure stability, we use special finite difference operators that satisfy a summation-by-parts (SBP) property, mimicking integration by parts for continuous functions. Physical boundary and interface conditions are imposed via simultaneous-approximation-term (SAT) technique. Non-conforming grid interface are handled by interface operators. The resulting schemes can be proven stable with the energy method.

When using the SBP-SAT technique to solve an initial boundary value problem, the truncation error is larger near the boundary than that in the interior of the computational domain. If the mesh is block structured, and different mesh refinements are used in the blocks, the large truncation error is near the edge of the interface. Typically, the large truncation error is localized in a low dimension space, and is expected to dominate the overall error of the numerical solution. However, many numerical experiments show that the convergence rate is not as low as expected. In this project, we analyze how the large localized truncation error affects the convergence of the numerical methods for the second order wave equation.

Because ordinary finite-difference methods become unwieldy in the presence of complicated geometry, recent efforts have been directed towards an immersed boundary technique. There, the grid points do not have to coincide with the boundary.

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An immersed finite difference grid. |
A simulation of the acoustic wave equation with the immersed finite difference technique. The initial contition at time t=0 was a Gaussian pulse centered at (x,y) = (0,0). |

#### Immersed Finite Elements

We investigate how to impose boundary and interface conditions with finite elements for problems when the mesh is not aligned with the domain. This would be referred to as immersed finite elements. Using an unfitted mesh typically leads to an ill-conditioned system when some elements have only a small part inside the computational domain. This can be remedied by additional penalty terms which are investigated in this project. Recent focus has been on weakly imposing boundary conditions using Nitsche's method together with appropriate penalty terms. So far the acoustic wave equation has been considered.

### Publications

- Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation
- A Stabilized Nitsche Cut Element Method for the Wave Equation
- High Order Finite Difference Methods for the Wave Equation with Non-conforming Grid Interfaces
- Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media
- Atmospheric Sound Propagation Over Large-scale Irregular Terrain

**Contact**: , Gunilla Kreiss