On L2-dissipativity of linearized explicit finite-difference schemes with a regularization on a non-uniform spatial mesh for the 1D gas dynamics equations

Conditions for $L^2$-dissipativity of an explicit symmetric finite-difference scheme for linearized 2d and 3d gas dynamics equations with a regularization

“…For an initial-boundary value problem on a non-uniform rectangular grid, sufficient Courant-type conditions for the L 2 -dissipativity and stability are given for the first time using the energy method. For the 1D equations, see [11] in the case of zero background velocity and the uniform spatial mesh and [16] in the general case, for the non-uniform spatial mesh and with stronger results; also see [20] concerning unconditional stability of implicit finite-difference schemes for related but another linearized multidimensional second order in time hyperbolic QGD regularization. To this end, in this paper, first the sufficient conditions for dissipativity and stability of abstract explicit two-level schemes with a non-self-adjoint operator in a Euclidean space are derived.…”

“…Consequently, here G m ≥ 0. 14) is not only sufficient but also necessary for the validity of property (16).…”