High-Order Finite Difference Methods, Multidimensional Linear Problems, and Curvilinear Coordinates

Boundary and interface conditions are derived for high-order finite difference methods applied to multidimensional linear problems in curvilinear coordinates. Difficulties presented by the combination of multiple dimensions and varying coefficients are analyzed. In particular, problems related to nondiagonal norms, a varying Jacobian, and varying and vanishing wave speeds are considered. The boundary and interface conditions lead to conservative schemes and strict and strong stability provided that certain metric conditions are met.