Stability analysis of the VC2 confinement scheme for the linear transport equation

An analysis of the stability of the VC2 confinement scheme applied to the linear transport equation is completed. In the general case, the energy method does not allow us to conclude, but it is found that the discrete energy of asymptotic solutions of the VC2 scheme cannot increase whatever the confinement parameter. Upper bounds on the magnitude of the confinement parameter are obtained by requiring that the confinement term do not change the sign of the solution. Such a constraint is mainly restrictive at the base of asymptotic pulse solutions, where a too large confinement parameter would generate a new pulse of opposite sign, which could grow without bounds together with the original one. Numerical tests indicate the relevance of this analysis.