High-Order, Finite-Volume Methods in Mapped Coordinates

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and/or face-centered values. Among the face averages to be computed arethose of the coordinate mapping metric factors, whose calculation affects notonly the overall accuracy of the scheme but also freestream preservation. Thelatter is automatically achieved to machine roundoff by representing the metricfactors as exterior derivatives, whose face averages are in turn reduced toquadratures on face hyperedges. The quadratures can be performed by anyconvenient method of sufficient accuracy.To demonstrate the approach, we developed fourth-order discretizations ofprototypical elliptic and hyperbolic problems. In addition to testing fourthorderaccuracy, the elliptic example displayed the use of an operator based ona second-order finite-volume discretization as a preconditioner in a conjugategradient iteration. Such strategies can be important in reducing the solver costof the larger stencils that inevitably accompany high-order discretizations. Inthe hyperbolic examples, we included an improved nonlinear limiter [8]. Ourresults indicate that the limited method can achieve a fourth-order convergencerate for smooth data on mapped grids and can control oscillations fordiscontinuous data on mapped grids.Although we focused on fourth-order, finite-volume discretizations in this paper,higher-order discretizations are obtainable by a similar strategy. The essentialingredients are a generalization of the cell face average product formula(17), more accurate quadrature formulas for the integrals of metric factors overface hyperedges in (11), and a discretization of flux face averages to the desiredorder of accuracy. Further extensions include moving mapped grids, as well asgeneralization from domains describable by a smooth mapping from a singleCartesian computational grid to more complicated multiblock geometries.References[1] M. Vinokur, An analysis of finite-difference and finite-volume formulations ofconservation laws, J. Comput. Phys. 81 (1989) 1–52.[2] D. Calhoun, R. J. LeVeque, An accuracy study of mesh refinement onmapped grids, in: T. Plewa (Ed.), Adaptive Mesh Refinement - TheoryAnd Applications: Proceedings of The Chicago Workshop On Adaptive MeshRefinement Methods, Vol. 41 of Lecture Notes in Computational Science andEngineering, Springer Verlag, 2003, pp. 91–102.[3] D. A. Calhoun, C. Helzel, R. J. Leveque, Logically rectangular grids and finitevolume methods for PDEs in circular and spherical domains, SIAM Rev. 50 (4)(2008) 723–752.[4] M. Barad, P. Colella, A fourth-order accurate local refinement method forPoisson’s equation, J. Comput. Phys. 209 (2005) 1–18.34
[5] P. D. Thomas, C. K. Lombard, Geometric conservation law and its applicationto flow computations on moving grids, AIAA J. 17 (10) (1979) 1030–1037.[6] D. A. Kopriva, Metric identities and the discontinuous spectral element methodon curvilinear meshes, SIAM J. Sci. Comput. 26 (3) (2006) 301–327.[7] M. R. Visbal, D. V. Gaitonde, On the use of high-order finite-difference schemeson curvilinear and deforming meshes, J. Comput. Phys. 181 (1) (2002) 155–185.[8] P. Colella, M. D. Sekora, A limiter for PPM that preserves accuracy at smoothextrema, J. Comput. Phys. 227 (2008) 7069–7076.[9] M. Spivak, Calculus on Manifolds, W. A. Benjamin, Inc., New York, NY, 1965.[10] C. H. Evans, J. H. Hawley, Simulation of magnetohydrodynamic flows - aconstrained transport method, Astrophysical Journal 332 (1988) 659.[11] J. B. Bell, P. Colella, J. A. Trangenstein, M. Welcome, Adaptive meshrefinement on moving quadrilateral grids, in: Proceedings, 9th AIAAComputational Fluid Dynamics Conference, Buffalo, NY, 1989, pp. 471–479,aIAA-1989-1979.[12] R. L. Miller, M. S. Chu, J. M. Greene, Y. R. Lin-Liu, R. E. Waltz, Noncircular,finite aspect ratio, local equilibrium model, Phys. Plasmas 5 (4) (1998) 973–978.[13] http://www.netlib.org/quadpack/.[14] P. Colella, D. T. Graves, T. J. Ligocki, D. F. Martin, D. Modiano, D. B.Serafini, B. V. Straalen, Chombo Software Package for AMR Applications -Design Document, http://seesar.lbl.gov/anag/chombo.[15] http://computation.llnl.gov/casc/hypre.[16] P. Colella, P. R. Woodward, The Piecewise Parabolic Method (PPM) for gasdynamicalsimulations, J. Comput. Phys. 54 (1) (1984) 174–201.[17] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, JohnWiley & Sons, New York, 1991.[18] R. D. Richtmyer, K. W. Morton, Difference Methods for Initial-Value Problems,2nd Edition, John Wiley & Sons, New York, 1967.[19] B. Gustafsson, H.-O. Kreiss, J. Oliger, Time Dependent Problems andDifference Methods, Pure and Applied Mathematics, John Wiley & Sons, NewYork, 1995.[20] P.-O. Persson, J. Bonet, J. Peraire, Discontinuous Galerkin solution of theNavier-Stokes equations on deformable domains, Comput. Methods Appl. Mech.Eng. 198 (2009) 1585–1595.35
Page 1 and 2: High-Order, Finite-Volume Methods i
Page 3 and 4: the exact enforcement of the metric
Page 5: If we define control volumes in phy
Page 8 and 9: Alternative expressions to (17) are
Page 10: β 1 i+2 ed +e❞ ❞ ❞ d′❞β
Page 13 and 14: ✠Dirichlet boundaryDCBAFig. 3. Lo
Page 15 and 16: 10.80.60.40.20−0.2−0.4−0.6−
Page 17: 10 −2 N10 −310 −410 −510
Page 21 and 22: 3Runge Kutta Stability Domain21Im z
Page 23 and 24: Magnitude of Amplification FactorRe
Page 26 and 27: L ∞ Error Norm for Base SchemeN c
Page 28 and 29: 10 0 Uniform Advection of Sinusoida
Page 30 and 31: Note that, in computational space,
Page 32 and 33: 10 0 Uniform Advection of Compact D

High-Order, Finite-Volume Methods in Mapped Coordinates

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