Energy stable high order finite difference methods for hyperbolic equations in moving coordinate systems

A time-dependent coordinate transformation of a constant coeficient hyperbolic equa- tion which results in a variable coeficient problem is considered. By using the energy method, we derive well-posed boundary conditions for the continuous problem. It is shown that the number of boundary conditions depend on the coordinate transformation. By using Summation-by-Parts (SBP) operators for the space discretization and weak boundary conditions, an energy stable finite difference scheme is obtained. We also show how to construct a time-dependent penalty formulation that automatically imposes the right number of boundary conditions. Numerical calculations corroborate the stability and accuracy of the approximations.