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(2011), No. 1, 1–25 Communications in Applied Mathematics and Computational Science vol. 6 no. 1 2011 mathematical sciences publishers Communications in Applied Mathematics and Computational Science pjm.math.berkeley.edu/camcos EDITORS MANAGING EDITOR John B. Bell Lawrence Berkeley National Laboratory, USA [email protected] BOARD OF EDITORS Marsha Berger New York University Ahmed Ghoniem Massachusetts Inst. of Technology, USA [email protected] [email protected] Alexandre Chorin University of California, Berkeley, USA Raz Kupferman The Hebrew University, Israel [email protected] [email protected] Phil Colella Lawrence Berkeley Nat. Lab., USA Randall J. LeVeque University of Washington, USA [email protected] [email protected] Peter Constantin University of Chicago, USA Mitchell Luskin University of Minnesota, USA [email protected] [email protected] Maksymilian Dryja Warsaw University, Poland Yvon Maday Université Pierre et Marie Curie, France [email protected] [email protected] M. Gregory Forest University of North Carolina, USA James Sethian University of California, Berkeley, USA [email protected] [email protected] Leslie Greengard New York University, USA Juan Luis Vázquez Universidad Autónoma de Madrid, Spain [email protected] [email protected] Rupert Klein Freie Universität Berlin, Germany Alfio Quarteroni Ecole Polytech. Féd. Lausanne, Switzerland [email protected] alfio.quarteroni@epfl.ch Nigel Goldenfeld University of Illinois, USA Eitan Tadmor University of Maryland, USA [email protected] [email protected] Denis Talay INRIA, France [email protected] PRODUCTION [email protected] Silvio Levy, Scientific Editor Sheila Newbery, Senior Production Editor See inside back cover or pjm.math.berkeley.edu/camcos for submission instructions. The subscription price for 2011 is US $70/year for the electronic version, and $100/year for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA. Communications in Applied Mathematics and Computational Science, at Mathematical Sciences Publishers, Department of Mathemat- ics, University of California, Berkeley, CA 94720-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. CAMCoS peer review and production are managed by EditFLOW™ from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers http://msp.org/ A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2011 by Mathematical Sciences Publishers COMM. APP. MATH. AND COMP. SCI. Vol. 6, No. 1, 2011 msp A HIGH-ORDER FINITE-VOLUME METHOD FOR CONSERVATION LAWS ON LOCALLY REFINED GRIDS PETER MCCORQUODALE AND PHILLIP COLELLA We present a fourth-order accurate finite-volume method for solving time-depen- dent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement. The underlying method is a generalization of that devel- oped by Colella, Dorr, Hittinger and Martin (2009) to nonlinear systems, and is based on using fourth-order accurate quadratures for computing fluxes on faces, combined with fourth-order accurate Runge–Kutta discretization in time. To interpolate boundary conditions at refinement boundaries, we interpolate in time in a manner consistent with the individual stages of the Runge–Kutta method, and interpolate in space by solving a least-squares problem over a neighborhood of each target cell for the coefficients of a cubic polynomial. The method also uses a variation on the extremum-preserving limiter of Colella and Sekora (2008), as well as slope flattening and a fourth-order accurate artificial viscosity for strong shocks. We show that the resulting method is fourth-order accurate for smooth solutions, and is robust in the presence of complex combinations of shocks and smooth flows. 1. High-order finite-volume methods In the finite-volume approach, the spatial domain in RD is discretized as a union of rectangular control volumes that covers the spatial domain. For Cartesian-grid finite-volume methods, a control volume Vi takes the form D Vi DTih;.i C u/hU for i 2 Z ; u D .1; 1;:::; 1/; where h is the grid spacing. A finite-volume discretization of a partial differential equation is based on averaging that equation over control volumes, applying the divergence theorem to replace volume integrals by integrals over the boundary of the control volume, This work was supported by the Director, Office of Science, Office of Advanced Scientific Computing Research, of the U.S. Department of Energy under contract no. DE-AC02-05CH11231. MSC2000: 65M55. Keywords: high-order methods, finite-volume methods, adaptive mesh refinement, hyperbolic partial differential equations. 1 2 PETERMCCORQUODALE AND PHILLIP COLELLA and approximating the boundary integrals by quadratures. In this paper, we solve time-dependent problems that take the form of a conservation equation: @U C r · FE.U/ D 0: (1) @t The discretized solution in space is the average of U over a control volume, 1 Z hUi t D U t d i . / D .x; / x: (2) h Vi We can compute the evolution of the spatially discretized system by a method- of-lines approach, h i Z d U i 1 E 1 P d d D − r · Fdx D − hF i C 1 d − hF i − 1 d ; (3) D i 2 e i 2 e dt h Vi h d Z d 1 d hF ii± 1 ed D F d A; (4) 2 h D−1 ± Ad ± where Ad are the high and low faces bounding Vi with normals pointing in the ed direction. In this case, the finite-volume approach computes the average of the divergence of the fluxes on the left side of (4) with the sum of the integrals over faces on the right side, with the latter approximated using some quadrature rule. Such approximations are desirable because they lead to conserved quantities in the original PDE satisfying an analogous conservation law in the discretized system. The approach we take in this paper is a generalization of the method in [5] to general nonlinear systems of hyperbolic conservation laws on locally refined grids, using fourth-order quadratures in space to evaluate the flux integrals (4) on the faces [1], and a Runge–Kutta method for evolving the ODE (3). We use this approach as the starting point for a block-structured adaptive mesh refinement method along the lines of that in [3]. 2. Single-level algorithm 2.1. Temporal discretization. Given the solution hUin ≈ hUi.tn/, we compute a fourth-order temporal update to hUinC1 ≈ hUi.tn C 1t/ using the classical fourth- order Runge–Kutta (RK4) scheme on (1). We are solving the autonomous system of ODEs dhUi D −D · FE; (5) dt E E 1 X d d D · F D D · F.hUi/ D hF iiC 1 ed − hF ii− 1 ed : h 2 2 d CONSERVATIONLAWSONLOCALLYREFINEDGRIDS 3 Then, starting with hUi.0/ D hUi.tn/, set E .0/ k1 D −D · F.hUi /1t; (6) h i.1/ D h i.0/ C 1 D − · E h i.1/ U U 2 k1; k2 D F. U /1t; (7) h i.2/ D h i.0/ C 1 D − · E h i.2/ U U 2 k2; k3 D F. U /1t; (8) .3/ .0/ E .3/ hUi D hUi C k3; k4 D −D · F.hUi /1t: (9) Then to integrate one time step: h i n C D h i n C 1 C C C C 5 U .t 1t/ U .t / 6 .k1 2k2 2k3 k4/ O..1t/ /: (10) The method given above is in conservation form. That is, h inC1 D h in − 1t Ph d itot − h d itot U U F iC 1 ed F i− 1 ed ; (11) h d 2 2 h d itot D 1 h d i.0/ C h d i.1/ C h d i.2/ C h d i.3/ F C 1 d F C 1 d 2 F C 1 d 2 F C 1 d F C 1 d ; i 2 e 6 i 2 e i 2 e i 2 e i 2 e d .s/ d .s/ h i D h h i i 1 F C 1 d F . U / iC ed : i 2 e 2 2.2. Spatial discretization. To complete the definition of the single-level algorithm, d we need to specify how to compute hF i C 1 d as a function of hUi. Our approach i 2 e generalizes that in [5] to the case of nonlinear systems of conservation laws. Fol- lowing what often is done for second-order methods, we introduce a nonlinear change of variables W D W.U/. In the case of gas dynamics, this is the conversion from the conserved quantities mass, momentum, and energy, U D (ρ; ρuE; ρE/, to primitive variables W D (ρ; uE; p/, where ρ is the gas density, uE is the velocity vector, E is the total energy per unit mass, and p is the pressure. Typically, this transformation is done to simplify the limiting process, for example, to permit the use of component-wise limiting. Some care is required in transforming from conservative to primitive variables in order to preserve fourth-order accuracy. 1. Convert from cell-averaged conserved variables to cell-averaged primitive variables, through cell-centered values, as follows. Calculate a fourth-order approximation to U at cell centers: h2 U D hUi − 1.2/hUi ; (12) i i 24 i where 1.2/ is the second-order accurate Laplacian .2/ X 1 1 q D .q d − 2q C q d /: (13) i h2 i−e i iCe d 4 PETERMCCORQUODALE AND PHILLIP COLELLA Then convert to primitive variables: Wi D W.Ui /; (14) W i D W.hUii /: (15) Calculate a fourth-order approximation to cell-averaged W: h i D C 1 2 .2/ W i Wi 24 h 1 W i : (16) 2.
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