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The multiplicative (or geometric) calculus is a non-Newtonian calculusderived from an arithmetic in which the operations ofaddition/subtraction/multiplication are replaced bymultiplication/division/exponentiation. A major difference between themultiplicative calculus and the classical additive calculus, and one that hasimportant consequences in the simulation of wave propagation problems, is thatin geometric calculus the role of polynomials is played by exponentials of apolynomial argument. For example, whereas a polynomial of degree one hasconstant (classical) derivative, it is the exponential function that hasconstant derivative in the multiplicative calculus. As we will show, thisimplies that even low-order finite quotient approximations|the analogues offinite differences in the multiplicative calculus|produce exact multiplicativederivatives of exponential functions. We exploit this fact to show that somepartial differential equations (PDE) can be solved far more efficiently usingtechniques based on the multiplicative calculus. For wave propagation models inparticular, we will show that it is possible to circumvent theminimum-points-per-wavelength sampling constraints of classical methods. Inthis first part we develop the theoretical framework for studyingmultiplicative partial differential equations and their connections withclassical models.

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