A multilevel Cartesian non-uniform grid time domain algorithm

Fast convolution quadrature for the wave equation in three dimensions

“…The double integral in (21) can be split into a sum of two double integrals: one over the leaves of the block-cluster tree belonging to the set L d (T I×I ) and the other being the remainder. Namely, , for all j ≥ L andd < d. (25) Recall that d is defined by (19).…”

“…The field of fast solvers that are based on MOT is relatively well-developed. Recent advances in this field include the seminal works [16,17] on the plane-wave time-domain algorithm, [18,19] on time-domain adaptive integral equation methods and [20,21] on the nonuniform (Cartesian) grid time-domain algorithms.…”

An efficient solution of time domain boundary integral equations for acoustic scattering and its acceleration by Graphics Processing Units

“…Another method for time domain March-On-in-Time acceleration is based on the observation that a specially constructed delay-and amplitude-compensated radiation field of a given group of sources can be represented by an extremely sparse and highly non-uniform space grid over the far-field. 4, 7, 8 A recent Cartesian Nonuniform Grid Time Domain Algorithm (CNGTDA) has been given in [40,47] exploiting the smoothness of the compensated field. This method will be used in the current study.…”

“…We note that the compensated field is to be constructed in the characteristics time τ , whose grid usually will not coincide with that of the real time t. The values at the source locations required for (47) can be obtained by interpolation. If the same temporal basis functions as (35) are used, we have, for any given t, q(r, t) = q(r, t n )Ψ(t − t n ) + q(r, t n−1 )Ψ(t − t n−1 ) + q(r, t n−2 )Ψ(t − t n−2 ) + q(r, t n−3 )Ψ(t − t n−3 ) (53)…”

“…15,22 The second is the multi-level Cartesian Non-uniform Grid Time Domain algorithm (CNGTDA) based on far-field smoothness of the wave equation kernel. 4,40,47 In both approaches, the computational complexity can be reduced formally from O(N t N 2 ) to as low as O(N t N log 2 N ), where N t is the number of time steps and N is the total number of unknowns. In the present paper, a new, and simplified, algorithm that exploits far field smoothness of the kernel is proposed.…”