Abstract
This note presents a preliminary study of the supra-convergence of well-balanced (finite volume) schemes for conservation laws with a (nonlinear) geometrical source term. In particular, we consider scalar (linear) advection equations in one dimension, for which smooth analytical solutions are available, and upwind interfacial discretizations for the numerical simulation on nonuniform grids (also generated by adaptive procedures). We point out the inconsistent characteristics of the (local) truncation error, and then we illustrate through a simple example the consistency condition formulated by Wendroff and White, which turns out to be effectively compatible with a (strong) convergence theory at optimal rates.
Similar content being viewed by others
References
Arvanitis, C.: Mesh redistribution strategies and finite element schemes for hyperbolic conservation laws. J. Sci. Comput. 34(1), 1–25 (2008)
Arvanitis, C., Delis, A.I.: Behavior of finite volume schemes for hyperbolic conservation laws on adaptive redistributed spatial grids. J. Sci. Comput. 28(5), 1927–1956 (2006)
Ben-Artzi, M., Falcovitz, J.: An upwind second-order scheme for compressible duct flows. SIAM J. Sci. Stat. Comput. 7(3), 744–768 (1986)
Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23(8), 1049–1071 (1994)
Bouche, D., Ghidaglia, J.-M., Pascal, F.: Error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation. SIAM J. Numer. Anal. 43(2), 578–603 (2005)
Bouchut, F., Perthame, B.: Kružkov’s estimates for scalar conservation laws revisited. Trans. Am. Math. Soc. 350(7), 2847–2870 (1998)
Cockburn, B., Gremaud, P.-A.: A priori error estimates for numerical methods for scalar conservation laws. II. Flux-splitting monotone schemes on irregular Cartesian grids. Math. Comp. 66(218), 547–572 (1997)
Després, B.: Lax theorem and finite volume schemes. Math. Comp. 73(247), 1203–1234 (2004)
Després, B.: An explicit a priori estimate for a finite volume approximation of linear advection on non-Cartesian grids. SIAM J. Numer. Anal. 42(2), 484–504 (2004)
Hoffman, J.D.: Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes. J. Comput. Phys. 46(3), 469–474 (1982)
Kreiss, H.-O., Manteuffel, T.A., Swartz, B., Wendroff, B., White, A.B., Jr.: Supra-convergent schemes on irregular grids. Math. Comp. 47(176), 537–554 (1986)
Perthame, B., Simeoni, C.: Convergence of the Upwind Interface Source method for hyperbolic conservation laws. In: Hyperbolic Problems: Theory, Numerics, Applications, pp. 61–78. Springer, Berlin (2003)
Pike, J.: Grid adaptive algorithms for the solution of the Euler equations on irregular grids. J. Comput. Phys. 71(1), 194–223 (1987)
Roe, P.L.: Upwind differencing schemes for hyperbolic conservation laws with source terms. In: Carasso, C., Raviart, P.A., Serre, D. (eds.) Nonlinear Hyperbolic Problems. Lecture Notes in Math., vol. 1270, pp. 41–51. Springer, Berlin (1987)
Rogers, B.D., Borthwick, A.G.L., Taylor, P.H.: Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J. Comput. Phys. 192(2), 422–451 (2003)
Sfakianakis, N.: Finite difference schemes on non-uniform meshes for Hyperbolic Conservation Laws. Ph.D. thesis, University of Crete, Heraklion (2009)
Teng, Z.-H.: Modified equation with adaptive monotone difference schemes and its convergent analysis. Math. Comp. 77(263), 1453–1465 (2008)
Tikhonov, A.N., Samarsky, A.A.: On the theory of homogeneous difference schemes. In: Outlines Joint Sympos. Partial Differential Equations, Novosibirsk, 1963, pp. 266–273. Acad. Sci. USSR Siberian Branch, Moscow (1963)
Turkel, E.: Accuracy of schemes with nonuniform meshes for compressible fluid flows. Appl. Numer. Math. 2(6), 529–550 (1986)
Vasilyev, O.V.: High order finite difference schemes on non-uniform meshes with good conservation properties. J. Comput. Phys. 157(2), 746–761 (2000)
Wendroff, B., White, A.B., Jr.: A supraconvergent scheme for nonlinear hyperbolic systems. Comput. Math. Appl. 18(8), 761–767 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Simeoni, C. Remarks on the Consistency of Upwind Source at Interface Schemes on Nonuniform Grids. J Sci Comput 48, 333–338 (2011). https://doi.org/10.1007/s10915-010-9442-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-010-9442-6