Numerical Solution of 3D Exterior Unsteady Wave Propagation Problems Using Boundary Operators

SIAM Journal of Scientific Computing. 2020. Vol. 42. No. 5. P. A3462–A3488.

Petropavlovsky S., Tsynkov S., Turkel E.

We propose a boundary method for the numerical simulation of time-dependent
waves in three-dimensional (3D) exterior regions. The order of accuracy can be either second or
fourth in both space and time. The method reduces a given initial boundary value problem for
the wave equation to a set of operator equations at the boundary of the original domain. The
reduction is based on a reformulation of the method of difference potentials. The resulting operator
equations relate the solution and its normal derivative at the boundary. To solve these equations,
one relies on the Huygens' principle. This yields an algorithm that works on a sliding time window
of a finite nonincreasing duration. As a result, it allows one to avoid the ever increasing backward
dependence of the solution on time. The major advantages of the proposed methodology are its
reduced computational complexity (grid-independent on the boundary and sublinear in the volume),
the capacity to handle curvilinear geometries using Cartesian finite difference time domain (FDTD)
methods, and automatic and exact accounting for the far-field radiation conditions. In addition, the
methodology facilitates solution of multiple similar problems al low individual cost per problem and
guarantees uniform performance over arbitrarily long time intervals.

Keywords: Method of difference potentials (MDP), метод разностных потенциалов, Huygens' principle, граничные операторы