SUMMARYAfter several years of planning, the 1st International Workshop on High‐Order CFD Methods was successfully held in Nashville, Tennessee, on January 7–8, 2012, just before the 50th Aerospace Sciences Meeting. The American Institute of Aeronautics and Astronautics, the Air Force Office of Scientific Research, and the German Aerospace Center provided much needed support, financial and moral. Over 70 participants from all over the world across the research spectrum of academia, government labs, and private industry attended the workshop. Many exciting results were presented. In this review article, the main motivation and major findings from the workshop are described. Pacing items requiring further effort are presented. Copyright © 2013 John Wiley & Sons, Ltd.

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Upwind methods for the 1-D Euler equations, such as TVD schemes based on Roc's approximate Riemann solver, are reinterpreted as residual distribution schemes, assuming continuous piecewise linear space variation of the unknowns defined at the cell vertices. From this analysis three distinct steps are identified, each allowing for a multidimensional generalization without reference to dimensional splitting or I-D Riemann problems. A key element is the necessity to have continuous piecewise linear variation of the unknowns, requiring linear triangles in two space dimensions and tetrahedra in three space dimensions. Flux differences naturally generalize to flux contour integrals over the triangles. Rce's flux difference splitter naturally generalizes to a multidimensional flux balance splitter if one assumes that the parameter vector variable is the primary dependent unknown having linear variation in space. Nonlinear positive and second-order scalar distribution schemes provide a true generalization of the TVD schemes in one space dimension. Although refinements and improvements are still possible for all these elements, computational examples show that these generalizations present a new framework for solving the multidimensional Euler equations.

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In this article, we introduce and analyze the class of numerical schemes known as residual distribution or fluctuation splitting schemes. The root of this family of discretizations is found mainly in works aiming at generalizing the finite volume techniques to a genuinely multidimensional upwinding context, as in, for example (Hall, Morton, Ni, Roe). However, the final result is a numerical method sharing a lot with other techniques, such as finite element schemes (Carette, Deconinck, Paillere and Roe, 1995). Our aim is to look at fluctuation splitting/residual distribution methods from a generic theoretical point of view. We discuss in a general way the basic principles and the criteria used in the design of the schemes: positivity, k‐exacteness, energy stability, well‐balancedness. We also show similarities with known, more classical, discretization techniques, as well as the main distinguishing features that make residual distribution methods a valid alternative to these classical schemes. Finally, we present some numerical applications and comparisons. Throughout the text, we give an extensive overview of the existing literature, and finally conclude the chapter by reviewing most of the ongoing research on the topic.

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Multidimensional upwind residual distribution schemes are extended to the context of continuous linear space-time ÿnite elements for the time accurate solution of scalar and hyperbolic systems of conservation laws. The formulation leads to a consistent discretization of the space-time domain, thus retaining the properties of the underlying basic schemes both in space and time. We propose a particular space -time mesh conÿguration containing two layers of elements and three levels of nodes in time. This construction leads to unconditionally stable implicit time stepping while retaining second-order spatial and temporal accuracy in smooth ows and monotone solution across steep gradients. The presented schemes have a strong potential in the ÿeld of moving grids, since they allow a dynamic change of the space-time mesh geometry. Numerical results demonstrate the robustness, accuracy and monotonicity of the method.