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Bram van Leer is Arthur B. Modine Emeritus Professor of aerospace engineering at the University of Michigan, in Ann Arbor.〔(【引用サイトリンク】van Leer at University of Michigan )〕 He specializes in ''Computational fluid dynamics (CFD)'', ''fluid dynamics'', and ''numerical analysis, '' fields on which he has had a substantial influence. An astrophysicist by education, Van Leer made seminal contributions to CFD in his 5-part article series “Towards the Ultimate Conservative Difference Scheme (1972-1979),” where he extended Godunov’s finite-volume scheme to the second order (MUSCL), developed nonoscillatory interpolation using limiters, an approximate Riemann solver, and Discontinuous-Galerkin schemes for unsteady advection. Since joining the University of Michigan’s Aerospace Engineering Department (1986) he has worked on convergence acceleration by local preconditioning and multigrid relaxation for Euler and Navier-Stokes problems, unsteady adaptive grids, space-environment modeling, atmospheric flow modeling, extended hydrodynamics for rarefied flows, and Discontinuous Galerkin methods. He retired in 2012. Throughout his career, van Leer has crossed interdisciplinary boundaries to export state-of-the-art CFD technology. Starting from astrophysics, he first made an impact on weapons research, followed by aeronautics, then space-weather modeling, atmospheric modeling, surface-water modeling and automotive engine modeling, to name the most important fields. ==Research work== Bram van Leer was a doctoral student in astrophysics at Leiden Observatory (1966-70) when he got interested in Computational Fluid Dynamics (CFD) for the sake of solving cosmic flow problems. His first major result in CFD was the formulation of the upwind numerical flux function for a hyperbolic system of conservation laws: Here the matrix appears for the first time in CFD, defined as the matrix that has the same eigenvectors as the flux Jacobian , but the corresponding eigenvalues are the moduli of those of . The subscript indicates a representative or average value on the interval ; it was no less than 10 years later before Philip L. Roe first presented his much used averaging formulas. Next, Van Leer succeeded in circumventing Godunov's barrier theorem (i.e., a monotonicity preserving advection scheme cannot be better than first-order accurate) by limiting the second-order term in the Lax-Wendroff scheme as a function of the non-smoothness of the numerical solution itself. This is a nonlinear technique even for a linear equation. Having discovered this basic principle, he planned a series of three articles titled "Towards the ultimate conservative difference scheme", which advanced from scalar nonconservative but non-oscillatory (part I) via scalar conservative non-oscillatory (part II) to conservative non-oscillatory Euler (part III). The finite-difference schemes for the Euler Equations turned out to be unattractive because of their many terms; a switch to the finite-volume formulation completely cleared this up and led to Part IV (finite-volume scalar) and, finally, Part V (finite-volume Lagrange and Euler) titled, "A second-order sequel to Godunov's method", which is his most cited (approaching 5000 citations in 2015) article. The series contains several original techniques that have found their way into the CFD community. In Part II two limiters are presented, later called by Van Leer "double minmod" (after Osher's "minmod" limiter) and its smoothed version "harmonic"; the latter limiter is sometimes referred to in the literature as "Van Leer's limiter." Part IV, "A new approach to numerical convection," describes a group of 6 second- and third-order schemes that includes two Discontinuous Galerkin schemes with exact time-integration. Van Leer was not the only one to break Godunov's barrier using nonlinear limiting; similar techniques were developed independently around the same time by Boris and by V.P. Kolgan, a Russian researcher unknown in the West. In 2011 Van Leer devoted an article to Kolgan's contributions and had Kolgan's 1972 TsADI report reprinted in translation in the Journal of Computational Physics. After the publication of the series (1972-79) Van Leer spent two years at ICASE (NASA LaRC), where he was engaged by NASA engineers interested in his numerical expertise. This led to Van Leer's differentiable flux-vector splittng 〔 〕 and the development of the block-structured codes CFL2D and CFL3D 〕〔 〕 which still are heavily used. Other seminal papers from these years are the review of upwind methods with Harten and Lax,〔 〕 the AMS workshop paper 〔 〕 detailing the differences and resemblances between upwind fluxes and Jameson's flux formula, and the conference paper with Mulder 〔 〕 on upwind relaxation methods; the latter includes the concept of Switched Evolution-Relaxation (SER) for automatically choosing the time-step in an implicit marching scheme. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bram van Leer」の詳細全文を読む スポンサード リンク
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