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High-Order Finite-Difference Schemes for Numerical Simulation of Hypersonic Boundary-Layer Transitionĭirect numerical simulation (DNS) has become a powerful tool in studying fundamental phenomena of laminar-turbulent transition of high-speed boundary layers.
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Results in two dimensions are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second- order algorithms. We prove discrete conservation for the new scheme and time stability in the absence of the flux correction terms. The resulting scheme possesses third- order design accuracy, but often exhibits fourth- order accuracy when higher- order derivatives are employed in the strand direction, especially for highly viscous flows. We show how this procedure allows for the proper truncation error canceling properties required for the flux correction scheme. Strand-direction derivatives are approximated to high-order via summation-by-parts operators for first derivatives and second derivatives with variable coefficients. The flux correction algorithm is applied in each unstructured layer of the strand grid, and the layers are then coupled together via a source term containing derivatives in the strand direction. High-order flux correction/ finite difference schemes for strand gridsĪ novel high-order method combining unstructured flux correction along body surfaces and high-order finite differences normal to surfaces is formulated for unsteady viscous flows on strand grids. A newly derived entropy stable weighted essentially non-oscillatory finite difference method is used to simulate problems with shocks and a conservative, entropy stable, narrow-stencil finite difference approach is used to approximate viscous terms. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equations.
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In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domainsĭeveloping stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis.