Explicit solutions of a convection-reaction equation and defects of difference schemes

We introduce two kinds of explicit solutions to the convection-reaction equation, ut + (|u| /q)x = u, u, x ∈ R, t ∈ R , q > 1, and employ them to test properties of various computational schemes. From this test we observe that computed solutions using Lax-Friedrichs, MacCormack and Lax-Wendroff schemes break down in a finite time. On the other hand some other schemes including WENO, NT and Godunov show more stable behavior and the tests provide their detailed behaviors. It is discussed that if a numerical scheme is applied to this problem together with the splitting method, certain defects of the scheme can be magnified exponentially and observed easily. Sometimes such a behavior destroys the numerical solution completely and hence one need to pay extra caution to deal with reaction dominant systems.