Explicit solutions to a convection-reaction equation and defects of numerical schemes

We develop a theoretical tool to examine the properties of numerical schemes for advection equations. To magnify the defects of a scheme we consider a convection-reaction equation u t + ( | u | q / q ) x = u , u , x ∈ R , t ∈ R + , q > 1 . It is shown that, if a numerical scheme for the advection part is performed with a splitting method, the intrinsic properties of the scheme are magnified and observed easily. From this test we observe that numerical solutions based on the Lax–Friedrichs, the MacCormack and the Lax–Wendroff break down easily. These quite unexpected results indicate that certain undesirable defects of a scheme may grow and destroy the numerical solution completely and hence one need to pay extra caution to deal with reaction dominant systems. On the other hand, some other schemes including WENO, NT and Godunov are more stable and one can obtain more detailed features of them using the test. This phenomenon is also similarly observed under other methods for the reaction part.