The objective of this work is to present the development of a numerical model for wave propagation in materials with time-varying, heterogeneous, and non-linear properties. Materials change with time as the result of complex linear and non-linear processes, which can occur due to natural causes or induced. Wave phenomena in this context brings about an interesting and complex problem, which involves the solution to coupled equations which describe interlinked multiphysics phenomena. Thus, understanding the dynamics of this interaction is beneficial to numerous applications across different industries and applied research; e.g. acoustic characterization of moving fluids, laser-fluid interaction, distributed optical fiber sensing, photonic integrated systems, among others. Numerical models, therefore, are indispensable to gain a deeper insight about the physical dynamics of the process and, ultimately, purvey a platform to design and test new applications and technologies.

Over time some numerical models have been proposed to simulate wave phenomena in these situations. The method and solution reviewed in this work provides a unique solution to develop and optimize multiple applications. For example, it can be used to model the interaction of electromagnetic waves with travelling Bragg mirrors produced by temperature or pressure changes in optical fibers, which is the basis of fiber-based distributed fiber sensing; the scattering of acoustic waves by transient disturbances in fluid flow that may arise from gas bubbles or variations in the density of fluids; and the propagation of an electromagnetic pulse in a rapidly moving and varying fluid.

The mathematical description of the process was derived originally for electromagnetics; yet, the numerical solver and mathematical treatment is generic and can be applied to other wave phenomena. The derivation departs from physical principles to write a generalized set of equations that describe wave propagation in time-varying, heterogeneous, and non-linear materials. The resulting set of hyperbolic partial differential equations (PDE) includes diffusive and convective terms that fully describe the wave interaction and process. Linear and nonlinear spatial and time heterogeneities in the material are assimilated into the convective terms of the hyperbolic wave equation. The solver was implemented using a semi-discrete and multidimensional scheme based in the finite-volume method which is highly scalable. Extension to other wave phenomena is discussed by analyzing the parameter correspondence for the acoustic and electromagnetic case.

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