Because direct simulation methods are, in fact, the subject of this book, it is worth saying a few words about the correspondence with the various other physical modeling methods discussed in the previous section. Indeed, after some exposure to these methods, it becomes clear that all can be related to one another, and to mainstream simulation methods.
Perhaps the closest relative of the direct techniques discussed here is the lumped mass-spring network methodology [38]; in some ways, this is more general than direct simulation approaches for distributed systems, in that one could design a lumped network without a distributed counterpart--this could indeed be attractive to a composer. As a numerical method however, it is designed as a large ordinary differential equation solver, which puts it in line with various simulation techniques based on semi-discretization, and in particular finite element methods. As mentioned in §1.2.1, distributed systems may be dealt with through large collections of lumped elements, and in this respect, the technique differs considerably from purely distributed models based on the direct solution of PDEs, because it can be quite cumbersome to design more sophisticated numerical methods, and to deal with systems more complex than a simple linear string or membrane using a lumped approach. The main problem is the ``local" nature of connections in such a network; in more modern simulation approaches, approximations at a given point in a distributed system are rarely modelled using nearest-neighbour connections between grid variables--see Chapter 13 for some examples. Still, it is possible to view the integration of lumped network systems in terms of distributed finite difference schemes--see §6.4.1 and §10.1.3 for details.
It should also come as no surprise that digital waveguide methods may also be rewritten as finite difference schemes. It is interesting that although the exact discrete traveling wave solution to the 1D wave equation has been known in the mainstream simulation literature for some time (since the 1960s at least [2]), and is a direct descendant of the method of characteristics [92], the efficiency advantage was apparently not taken advantage of to the same spectacular effect as in musical sound synthesis. (This is perhaps because the 1D wave equation is seen, in the mainstream world, as a model problem, and not of inherent practical interest.) Equivalences between finite differences and digital waveguide methods, in the 1D case and the multidimensional case of the waveguide mesh, have been established by various authors [235,237,209,208,22,74,192], and, as mentioned earlier, those at work on scattering based modular synthesis have incorporated ideas from finite difference schemes into their strategy [118,119]. This correspondence will be revisited with regard to the 1D wave equation in §6.2.9, and the 2D wave equation in §10.1.2. It is worth noting that the efficiency advantage of the digital waveguide method with respect to an equivalent finite difference scheme does not carry over to the multidimensional case [208,22].
Modal analysis and synthesis was in extensive use long before it appeared in musical sound synthesis applications, particularly in association with finite element analysis of vibrating structures--see [152] for an overview. In essence, a time-dependent problem, under some conditions, may be reduced to an eigenvalue, or statics problem, greatly simplifying analysis. It may also be viewed under the umbrella of more modern so-called spectral or pseudo-spectral methods [41]. Spectral methods essentially yield highly accurate numerical approximations through the use of various types of function approximations to the desired solution; many different varieties exist. If the solution is expressed in terms of trigonometric functions, the method is often referred to as a Galerkin Fourier method--this is exactly modal synthesis in the current context. Other types of spectral methods, perhaps more appropriate for sound synthesis purposes (and in particular collocation methods) will be discussed in Chapter 13. Modal synthesis methods will be discussed in more detail in §6.4.2 and §10.1.4.
Modular or ``hybrid" methods, though nearly always framed in terms of the language of signal processing may also be viewed as finite difference methods; the correspondence between lumped models and finite difference methods is direct, and that between wave digital filters and numerical integration formulae has been known for many years [84], and may be related directly to the even older concept of A-stability [94,63]. The key feature of modularity, however, is new to this field, and is not something which has been explored in depth in the mainstream simulation community.
This is not the place to evaluate the relative merit of the various physical modeling synthesis methods; this will be performed exhaustively with regard to two useful model problems, the 1D and 2D wave equations, in Chapters 6 and 10, respectively. For the impatient reader, some concluding remarks on relative strengths and weaknesses of these methods with respect to direct simulation methods appear in Chapter 14.