Overlapping Grid Method for Solving the Partial Differential Equations Using the SBP-SAT Technique

A SBP-SAT method with high time-stability of partial differential equations is derived that naturally introduce overlapping grid based on the Hermite interpolation. Derivative approximations that satisfy the summation by parts property and the boundary conditions are utilized using a penalty technique, simultaneous approximation terms, to guarantee the summation by parts property of the system. Time-stability is proven using the energy method and the finite difference methods based on this theory, which can establish the scheme that has incomparable advantages for simulating geometric discontinuity or complex media. The results show that the overlapping grid method of SBP-SAT methodology has better data transmission on the boundary and higher overall stability of the system.