Modelling wave propagation without sampling restrictions using the multiplicative calculus I: Theoretical considerations

The multiplicative (or geometric) calculus is a non-Newtonian calculus derived from an arithmetic in which the operations of addition/subtraction/multiplication are replaced by multiplication/division/exponentiation. A major difference between the multiplicative calculus and the classical additive calculus, and one that has important consequences in the simulation of wave propagation problems, is that in geometric calculus the role of polynomials is played by exponentials of a polynomial argument. For example, whereas a polynomial of degree one has constant (classical) derivative, it is the exponential function that has constant derivative in the multiplicative calculus. As we will show, this implies that even low-order finite quotient approximations|the analogues of finite differences in the multiplicative calculus|produce exact multiplicative derivatives of exponential functions. We exploit this fact to show that some partial differential equations (PDE) can be solved far more efficiently using techniques based on the multiplicative calculus. For wave propagation models in particular, we will show that it is possible to circumvent the minimum-points-per-wavelength sampling constraints of classical methods. In this first part we develop the theoretical framework for studying multiplicative partial differential equations and their connections with classical models.