Licentiate thesis 2010-004

Perfectly Matched Layers for Second Order Wave Equations

Kenneth Duru

7 May 2010


Numerical simulation of propagating waves in unbounded spatial domains is a challenge common to many branches of engineering and applied mathematics. Perfectly matched layers (PML) are a novel technique for simulating the absorption of waves in open domains. The equations modeling the dynamics of phenomena of interest are usually posed as differential equations (or integral equations) which must be solved at every time instant. In many application areas like general relativity, seismology and acoustics, the underlying equations are systems of second order hyperbolic partial differential equations. In numerical treatment of such problems, the equations are often rewritten as first order systems and are solved in this form. For this reason, many existing PML models have been developed for first order systems. In several studies, it has been reported that there are drawbacks with rewriting second order systems into first order systems before numerical solutions are obtained. While the theory and numerical methods for first order systems are well developed, numerical techniques to solve second order hyperbolic systems is an on-going research.

In the first part of this thesis, we construct PML equations for systems of second order hyperbolic partial differential equations in two space dimensions, focusing on the equations of linear elasto-dynamics. One advantage of this approach is that we can choose auxiliary variables such that the PML is strongly hyperbolic, thus strongly well-posed. The second is that it requires less auxiliary variables as compared to existing first order formulations. However, in continuum the stability of both first order and second order formulations are linearly equivalent. A turning point is in numerical approximations. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability for any given resolution. The second order discretization behaves much more stably. In the second order setting instability occurs only if unstable modes are well resolved.

The second part of this thesis discusses the construction of PML equations for the time-dependent Schrödinger equation. From mathematical perspective, the Schrödinger equation is unique, in the sense that it is only first order in time but second order in space. However, with slight modifications, we carry over our ideas from the hyperbolic systems to the Schrödinger equations and derive a set of asymptotically stable PML equations. The new model can be viewed as a modified complex absorbing potential (CAP). The PML model can easily be adapted to existing codes developed for CAP by accurately discretizing the auxiliary variables and appending them accordingly. Numerical experiments are presented illustrating the accuracy and absorption properties of the new PML model.

We are hopeful that the results obtained in this thesis will find useful applications in time-dependent wave scattering calculations.

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