constructhigh-fidelity Godunov finite volume
主办单位：北京应用物理与计算数学研究所
上海大学力学与工程科学学院
A new framework to constructhigh-fidelity Godunov finite volume schemes for both smooth and discontinuous solutions
肖锋
东京工业大学
Department of MechanicalEngineering
Tokyo Institute of Technology
(xiao.f.aa@m.titech.ac.jp)
Abstract：Godunov finite volume method (FVM) has been evolving into themainstream approach to compute numerical solutions of hyperbolic conservationlaws, including the Euler equations for compressible fluid dynamics. Spatialreconstruction, as the most essential part in an FVM, has been extensivelyinvestigated in the past decades. Varying among different discretizationtechniques though, the current paradigm to construct high-order FV schemes ismainly based on the following two steps: (i) using high-order polynomials,which can be built with either more DOFs over a wider stencil in theconventional FVM, or cell-wiselyincreased DOFs in discontinuous Galerkin,spectral element, or flux reconstruction methods; (ii) using nonlinear limitingprojection or artificial viscosity to “flatten” the reconstructed solution soas to suppress spurious oscillation in presence of large jumps ordiscontinuities. In spite of tremendoussuccess, the intrinsic inconsistency between (i) and (ii) usually puts a user ina dilemma to struggle simultaneously against numerical oscillation andexcessive dissipation, as evidenced in many applications.
In this talk, we present a novel numericalframework to accurately capture both smooth and discontinuous flow structures.The underlying concept is simple and straightforward, i.e. (1) prepare multiplereconstruction functions, so-called admissible functions, as the possiblecandidate functions to mimic smooth and discontinuous solutions respectively;and (2) devise an algorithm to select the best suited one out from thecandidate functions for spatial reconstruction. Motivated by the observationthat minimizing the jumps of the reconstructed physical variables at cellboundaries (BV) can effectively reduce the dissipation errors, we have recentlyproposed and practiced the Boundary Variation Diminishing (BVD) principle toderive the algorithm for (2). With some properly chosen admissiblereconstruction functions, we have developed a class of schemes of greatpractical significance for compressible flows involving both smooth anddiscontinuous solutions. Benchmark tests show that the present method cansimulate single and multiple phase compressible flows with substantiallyimproved solution quality in comparison to other existing methods.
Short Bio: Dr. Xiao is a professor in the department of mechanical engineering atTokyo Institute of Technology (Tokyo Tech). He is currently engaged inresearches on computational fluid dynamics and data/numerical-model integratedanalysis, he has authored or co-authored over 140 papers in academic journals.He is a fellow of Japan Society of Mechanical Engineers (JSME), a recipient ofJSME Computational Mechanics Achievement Award, JACM Computational MechanicsAward, JACM Fellow Award and others. Prof. Xiao is currently serving theexecutive editor of Journal of Computational Physics (JCP).
