On Enlarged Sufficient Conditions for $$L^2$$-Dissipativity of Linearized Explicit Schemes with Regularization for 1D Gas Dynamics Systems of Equations

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

“…then the operator G = c * B + c 2 * τ A satisfies inequality (17). Consequently, for the solution to the explicit scheme (19), bound (15) and, for b = 0, the H h -dissipativity property (16) are valid.…”

“…suf (α) increases on (0, α opt ], decreases on [α opt , +∞) and β (0) suf (α) → 0 as α → 0 + and α → +∞. Alternatively, one can try the choice τ = a ĥt /2 with the parameter a > 0 considered in [17]. Then condition ( 22) is transformed to the Courant-type form…”

“…For the Cauchy problem on a uniform mesh, both necessary conditions and sufficient conditions for the L 2 -dissipativity and stability, derived recently by the spectral method in [23], are improved. For the 1D case, see also [22] and, for enlarged sufficient conditions (based on better bounds for the commutator of convective and viscous terms), [17]. Also the simpler 2D and 3D barotropic case has recently been studied in [18].…”