We derive an a priori error estimate for the numerical solution obtained by time and space discretization by the finite volume/finite element method of the barotropic Navier-Stokes equations. The numerical solution on a convenient polyhedral domain approximating a sufficiently smooth bounded domain is compared with an exact solution of the barotropic Navier-Stokes equations with a bounded density. The result is unconditional in the sense that there are no assumed bounds on the numerical solution. It is obtained by the combination of discrete relative energy inequality derived in [17] and several recent results in the theory of compressible Navier-Stokes equations concerning blow up criterion established in [26] and weak strong uniqueness principle established in [10]. 1 p(z) z 2 dz.(2.4)Here and hereafter the symbolwhere T M > 0 is finite or infinite and depends on the initial data. Moreover, for any T *Step 2 By virtue of the weak-strong uniqueness result stated in [10, Theorem 4.1] (see also [14, Theorem 4.6]), the weak solution r, V coincides on the time interval [0, T M ) with the strong solution, the existence
Motivated by the work of Karper [29], we propose a numerical scheme to compressible Navier-Stokes system in spatial multi-dimension based on finite differences. The backward Euler method is applied for the time discretization, while a staggered grid, with continuity and momentum equations on different grids, is used in space. The existence of a solution to the implicit nonlinear scheme, strictly positivity of the numerical density, stability and consistency of the method for the whole range of physically relevant adiabatic exponents are proved. The theoretical part is complemented by computational results that are performed in two spatial dimensions.
We propose a mixed finite volume-finite element numerical method for solving the full Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat conducting fluid. The physical domain occupied by the fluid has a smooth boundary and it is approximated by a family of polyhedral numerical domains. Convergence and stability of the numerical scheme is studied. The numerical solutions are shown to converge, up to a subsequence, to a weak solution of the problem posed on the limit domain.
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