Simulation of Viscosity-Stratified Flow

Sammanfattning: The aim of this project is to study the viscous Burgers' equation for the case where the viscosity is constant, but also when it contains a jump in viscosity. In the first case where the viscosity is constant, Burgers' is simply solved on a singular domain. For the case with jump in viscosity, Burgers' is solved on multiple domains with different viscosity. The different domains are then connected by applying inner boundary conditions at an interface in order to produce a singular solution. The inner boundary conditions are imposed using three different methods; simultaneous approximation term (SAT), projection and hybrid method, where the hybrid method is a combination of both the SAT and projection method. These methods are used in combination with a stable and high-order accurate summation by parts (SBP) finite difference approximation in MATLAB. The three methods are then compared to each other with respect to the least square error and the corresponding convergence rate to determine which method is the most preferable to use. The methods resulting in the highest convergence rates are the projection and the hybrid methods. These methods manage to live up to the expected convergence rates for all operators with different orders of accuracy and are therefore both good methods to use. However, the best method to use is the projection method since it is much simpler to implement than the two other methods but still achieves just as good convergence rates as the hybrid method.