Revisiting Implicit Finite Difference Schemes for Three-Dimensional Room Acoustics Simulations

Implicit finite difference schemes for the 3-D wave equation using
a 27-point stencil on the cubic grid are presented, for use in room
acoustics modelling and artificial reverberation. The system of
equations that arises from the implicit formulation is solved using
the Jacobi iterative method. Numerical dispersion is analysed
and computational efficiency is compared to second-order accurate 27-point explicit schemes. Timing results from GPU implementations demonstrate that the proposed algorithms scale over their explicit counterparts as expected: by a factor of M + 2, where M is a fixed number of Jacobi iterations (eight can be sufficient in single precision). Thus, the accuracy of the approximation can be improved over explicit counterparts with only a linear increase in computational costs, rather than the quartic (in operations) and cubic (in memory) increases incurred when oversampling the grid. These implicit schemes are advantageous in situations where less than 1% dispersion error is desired.