Higher order semi-implicit schemes for linear advection equation on Cartesian grids with numerical stability analysis

A new class of semi-implicit numerical schemes for linear advection equation on Cartesian grids is derived that is inspired by so-called $κ$-schemes used with fully explicit discretizations for this type of problems. Opposite to fully explicit $κ$-scheme the semi-implicit variant is unconditionally stable in one-dimensional case and it preserves second order accuracy for dimension by dimension extension in higher dimensional cases. We discuss von Neumann stability conditions numerically for all numerical schemes. Using so-called Corner Transport Upwind extension of two-dimensional semi-implicit scheme with a special choice of $κ$ parameters, a second order accurate method is obtained for which numerical unconditional stability can be shown for variable velocity and the third order accuracy can be proved for constant velocity. Several numerical experiments illustrate the properties of semi-implicit schemes for chosen examples.