A stable boundary treatment of the dynamic beam equation (DBE) with two different sets of boundary conditions has been conducted using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) method. As the DBE involves a fourth derivative in space the numerical boundary treatment is highly non-trivial. Using SBP-SAT operators together with suitable time integration schemes the DBE has been simulated and a convergence study has been made. The results show that the SBP-SAT method produces a stable discretistation that is accurate enough to capture the dispersive nature of the dynamic beam equation. In additions simulations were made presenting the importance of a stable boundary treatment showing that the numerical solutions diverge when the boundaries were not handled correctly.

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.

This project studied two different ways of imposing boundary conditions weakly with the finite difference summation-by-parts (SBP) operators. These operators were combined with the boundary handling methods of simultaneous-approximation-terms (SAT) and the Projection to impose homogeneous Neumann and Dirichlet boundary conditions. The convergence rate of both methods was analyzed for different boundary conditions for the one-dimensional (1D) Schrödinger equation, without potential, which resulted in both methods performing similarly. A multi-block discretization was then implemented and different combinations of SBP-SAT and SBP-Projection were applied to impose inner boundary conditions of continuity between the blocks. A convergence study of the different methods of imposing the inner BC:s was conducted for the 1D Schrödinger equation without potential. The resulting convergence was the same for all methods and it was concluded that they performed similarly. Methods involving SBP-Projection had the slight advantage of faster computation time. Finally, the 1D Gross-Pitaevskii equation (GPE) and the 1D Schrödinger equation were analyzed with a step potential. The waves propagating towards the potential barrier were in both cases partially transmitted and partially reflected. The waves simulated with the Schrödinger equation dispersed, while the solitons simulated with the GPE kept their shape due to the equations reinforcing non-linear term. The bright soliton was partly transmitted and partly reflected. The dark soliton was either totally reflected or totally transmitted.