On the design of high order residual-based dissipation for unsteady compressible flows

The numerical dissipation operator of residual-based compact (RBC) schemes of high accuracy is identified and analysed for hyperbolic systems of conservation laws. A necessary and sufficient condition (χ-criterion) is found that ensures dissipation in 2-D and 3-D for any order of the RBC scheme. Numerical applications of RBC schemes of order 3, 5 and 7 to a diagonal wave advection and to a converging cylindrical shock problem confirm the theoretical results.