In this Bachelor project we researched the applicability of finite difference methods to solve the multi dimensional Westervelt equation. The Westervelt is an equation describing the propagation of a non-linear wave. The goal of this project was to solve the full Westervelt equation in higher dimensions. We used Julia for our implementations of the finite difference method. The choice of Julia allowed us to easily write high performing code. We were unsuccessful in modelling the full Westervelt equation since we were unable to implement attenuation into our models. We successfully implemented the non-linear wave equation without attenuation for a number of models, including a three dimensional model. The inability to implement attenuation into our models hampered the flexibility of our models. When the non-linearity created a shock front, the top of the wave would
experience numerical errors which quickly made the whole solution unphysical and ultimately unstable. An added implementation of attenuation would dampen the leading edge of the wave preventing it from becoming too steep. Additionally we made two comparisons with the linear wave equation to verify the correctness of our implementations.
We used a uniform grid to discretize the spatial dimensions. After which we used non-stiff FDM solvers with adaptive time stepping algorithms to solve for the remaining time dimension. We analyzed the use of stiff FDM solvers but these proved too computationally expensive to be viable, especially for our 3D model with 64.000 points.