Percussion Instrument Modelling In 3D: Sound Synthesis Through Time Domain Numerical Simulation
DOWNLOAD the PDF file (61 Mb)
Ap. B - Finite precision effects in FDTD schemes
This appendix offers novel insights on some round off
effects that appear in finite difference schemes. First, it is clarified what is
meant by conservation of energy "to machine precision", and how energy
variations should be calculated in order to highlight the correct quantisation.
This is an original contribution of this work. Then, the implementation of a
numerical scheme for the simple harmonic oscillator is analysed, both in the
lossless and lossy case. A second order implementation is compared with the
corresponding first order system. It is found that the first order system
behaves better than the second order implementation from an energy conservation
point of view, and that some numerical errors can be avoided with a different
implementation. The case of a free particle is analysed after. The second order
implementation and the first order system are compared numerically, and an
explanation for the quantised jumps that can be seen in the second order energy
plot is proposed. Finally, round off errors are analysed in the case of the 3D
wave equation. It is demonstrated how a naive implementation can lead to a
linear drift in the energy calculation, and how it is possible to cure this
artefact.