Applied Math Seminar

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Infinity in a Nullshell: Solving Wave Equations on Unbounded Domains

Speaker: Anil Zenginoglu, U Maryland

Location: Warren Weaver Hall 1302

Date: Friday, October 3, 2025, 2:30 p.m.

Synopsis:

When wave propagation problems are posed on unbounded domains, most numerical solvers rely on truncation, such as absorbing boundary conditions or perfectly matched layers (PML). I will describe a geometric alternative that solves the original exterior problem numerically: spatial compactification combined with a time shift that places infinity as a characteristic (null) boundary on the computational grid. This method links ideas between Lorentzian geometry and hyperbolic PDEs.

In the frequency domain, the time shift acts as a rephasing that keeps oscillations bounded so the effective wavenumber remains finite after compactification. In the time domain, the transformed first-order system is symmetric hyperbolic with maximally dissipative boundaries, ensuring stability and tracking energy decay.

For practical applications, a Null Infinity Layer (NIL) can wrap any interior mesh by a layer of finite thickness representing the unbounded exterior domain. I will present numerical experiments of benchmark scattering problems showing that NIL has comparable accuracy to PML while providing acces to the far-field.