Analysis and higher-order extension of the VC2 confinement scheme

Abstract This paper presents a detailed analysis of the VC2 confinement scheme for the linear transport equation. The confinement appears as a nonlinear second-order difference which drives the solution towards stable pulses when it operates in a globally anti-diffusive manner. Using the equivalent partial differential equation approach, both the confinement and the numerical dissipation of the discretization introduce forcing terms into the original equation, which govern the behavior of the solution for very large times. In a frame moving at the convection speed of the underlying equation, the signal is found to relax towards an asymptotic solution satisfying a fixed-point nonlinear difference equation expressing the balance between compression and diffusion. Based on the same ideas, a generalization of the VC2 confinement scheme to higher-orders is derived. It works for any odd-order scheme and is obtained by using a nonlinear mean between a centered and an upwind estimate of the lower-order dissipative term of the scheme. The asymptotic solution is the same for all orders of discretization and depends on the confinement scheme only. The solution of the transport equation relaxes towards this asymptotic solution at a rate depending on the order of the scheme. Numerical tests for various sets of initial conditions support the above findings.