A method of boundary equations for unsteady hyperbolic problems in 3D

Journal of Computational Physics. 2018. Vol. 365. P. 294–323.

Петропавловский С. В., Tsynkov S., Turkel E.

We consider interior and exterior initial boundary value problems for the three-dimensional
wave (d’Alembert) equation. First, we reduce a given problem to an equivalent operator
equation with respect to unknown sources defined only at the boundary of the original
domain. In doing so, the Huygens’ principle enables us to obtain the operator equation
in a form that involves only finite and non-increasing pre-history of the solution in time.
Next, we discretize the resulting boundary equation and solve it efficiently by the method
of  difference  potentials  (MDP).  The  overall  numerical  algorithm  handles  boundaries  of
general shape using regular structured grids with no deterioration of accuracy. For long
simulation times it offers sub-linear complexity with respect to the grid dimension, i.e., is
asymptotically cheaper than the cost of a typical explicit scheme. In addition, our algorithm
allows one to share the computational cost between multiple similar problems. On multi-
processor  (multi-core)  platforms,  it  benefits  from  what  can  be  considered  an  effective
parallelization in time.

Ключевые слова: The Huygens’ principle, Method of difference potentials (MDP), Calderon’s boundary equation, Parallelization in time