Compact Implicit MacCormack-Type Schemes with High Accuracy

In this work, the MacCormack methodology is extended to implicit compact differencing schemes. A prefactorization method is developed which splits the implicit matrix into two independent upper and lower matrices which are easier to invert. Using this method, a new class of high-order accurate compact MacCormack-type schemes is derived. Two fourth-order schemes are described, and results are shown for three linear and nonlinear CAA benchmark problems.