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On $$L^2$$-dissipativity of linearized explicit finite-difference schemes with a regularization on a non-uniform spatial mesh for the 1D gas dynamics equations. (English) Zbl 1411.65118
Summary: We deal with an explicit finite-difference scheme with a regularization for the 1D gas dynamics equations linearized at the constant solution. The sufficient condition on the Courant number for the $$L^2$$-dissipativity of the scheme is derived in the case of the Cauchy problem and a non-uniform spatial mesh. The energy-type technique is developed to this end, and the proof is both short and under clear conditions on matrices of the convective and regularizing (dissipative) terms. A scheme with a kinetically motivated regularization is considered as an application in more detail.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76M20 Finite difference methods applied to problems in fluid mechanics 76N15 Gas dynamics, general 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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