High Order Weighted Compact Boundary Condition

In multi-dimension flows, we expect to have problems at the boundaries when a shock hits or reflects at the boundary wall the remedy to this would be to develop weighted boundary conditions similar to the Weighted Compact Scheme for the interior nodes, by choosing candidate stencils around the boundary nodes and assigning weights to each of these stencils with the ENO reconstruction. This would avoid spurious oscillations when shocks are encountered at the boundaries. This thesis investigates higher order weighted compact boundary conditions for Weighted Compact Scheme (WCS). WCS is a combination of Essentially Non Oscillatory Scheme and Weighted Compact Finite Difference Schemes with Spectral-like Resolution. Implicit higher order schemes for spatial derivatives are derived for nodes in the neighborhood of the boundaries for the existing WCS scheme, which is used for the interior nodes. The objective is to achieve a higher weighted algorithm at the boundary, by using a compact stencil. To obtain this target, a combination of the spatial nodes' derivatives and values is going to be used. Several higher order schemes for the boundaries are derived and tested for both sine function and exponential function under 1st, 2nd and 3rd Boundary Condition. This new boundary scheme not only preserves the characteristic of standard compact schemes and achieves high order accuracy and high resolution using compact stencils, but also has the potential ability to accurately capture shock waves and discontinuities without oscillation. Numerical examples show the scheme is very promising and successful.