Numerical Simulation of Nonlinear and Dispersive Wave Equations using Adaptive Mesh Refinement (AMR)

Abstract

Numerical solution of time dependent Partial Differential Equations plays an important role in different fluid flow modelling problems. Sometimes a little portion of the computational domain needs high grid resolution in order to resolve phenomena such as steep fronts or shocks while use of a very high resolution mesh for the whole computational domain is a waste of computational resources since they are not required all over the domain. An Adaptive Mesh Refinement (AMR) procedure is an efficient and practical method for the numerical solution of Partial Differential Equation problems with regions of large gradients occupying a small subregion of the domain. An AMR algorithm refines grids by placing finer and finer subgrids in the different portions of the computational domain where they are required. For the time dependent problem the refinement is dynamic since the regions requiring refinement change with time and the AMR algorithm adaptively changes that. In this thesis we developed an AMR code for the numerical solution of linear, nonlinear and dispersive wave equations inspired by existing algorithms in the literature. In this work we kept the implementation simple and we use simple refinement criteria although the code allows for the use of more complex refinement criteria. In addition the implementation of the data structure was also kept simple. We have done the refinement in both time and space. In our code we generate finer grids which can also have finer grids using a recursive grid generation procedure. We give a review of some existing work along with the necessary components of our work. Numerical simulations of the linear advection equation, Burger's equation and the Regularized Long Wave (RLW) equation have been run with our AMR code. The results of these simulations are shown to have good agreement with numerical solutions obtained on fine resolution single grids which signify the success of our code. A significant time reduction in all the numerical simulations suggests the good performance of our code.