On differentiable local bounds preserving stabilization for Euler equations
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This work is focused on the design of nonlinear stabilization techniques for the finite element approximation of the Euler equations in steady form and the implicit time integration of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. We attempt to mitigate this issue by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. The importance of the differentiability is assessed by comparing the developed scheme with its non-differentiable counterpart. Numerical experiments suggest that, up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. In the case of transient problem, we also observe a reduction in the computational cost.
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