A novel 1D-FDTD scheme to solve the nonlinear second-order thermoviscous hydrodynamic model

Abstract

In this paper, we present a novel and simple Yee Finite-Difference Time-Domain (FDTD) scheme to solve numerically the nonlinear second-order thermoviscous Navier–Stokes and the Continuity equations. In their original form, these equations cannot be discretized by using the Yee’s mesh, at least, easily. As it is known, the use of the Yee’s mesh is recommended because it is optimized in order to obtain higher computational performance and remains at the core of many current acoustic FDTD softwares. In order to use the Yee’s mesh, we propose to rewrite the aforementioned equations in a novel form. To achieve this, we will use the substitution corollary. This procedure is novel in the literature. Although the scheme can be extended to more than one dimension, in this paper, we will focus only on the one-dimensional solution because it can be validated with two analytical solutions to the Burgers equation: the Mendousse mono-frequency solution and the Lardner bi-frequency solution. Numerical solutions are excellently consistent with the analytical solution, which demonstrates the effectiveness of our formulation.