A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions
The authors present a new time discretization method for retarded potential equations which circumvents numerical integration over intersections of the light cone with the spatial surface mesh. The method is tested on the wave equation on a sphere with Dirichlet boundary conditions on the boundary \(\Gamma\). They employ an ansatz as a single layer potential for the solution \(u\) of the wave equation: \[ u(x, t) := S\phi(x, t) :=\int_\Gamma \frac{\phi(y, t-\|x-y\|)}{4\pi\|x-y\|}\,d\Gamma_y, \] for \((x, t) \in \Omega\times[0,T]\), with unknown density function \(\phi\). The ansatz satisfies the wave equation and the initial conditions. Since the single layer potential can be extended continuously to the boundary \(\Gamma\), the unknown density function \(\phi\) is determined such that the boundary conditions \(u=g\) on \(\Gamma\times[0,T]\) are satisfied. This results in the boundary integral equation for \(\phi\), \(S\phi(x,t)=g(x,t)\) for all \((x,t)\in \Gamma\times[0,T]\), which is solved numerically by a space-time variational formulation.
scientific article; zbMATH DE number 6138628
scientific article