zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. (English) Zbl 1110.65077
Summary: In this article we present a non-oscillatory finite volume scheme of arbitrary accuracy in space and time for solving linear hyperbolic systems on unstructured grids in two and three space dimensions using the arbitrary high order schemes using derivatives (ADER) approach. The key point is a new reconstruction operator that makes use of techniques developed originally in the discontinuous Galerkin finite element framework. First, we use a hierarchical orthogonal basis to perform reconstruction. Second, reconstruction is not done in physical coordinates, but in a reference coordinate system which eliminates scaling effects and thus avoids ill-conditioned reconstruction matrices. In order to achieve non-oscillatory properties, we propose a new weighted essential nonoscillatory (WENO) reconstruction technique that does not reconstruct point-values but entire polynomials which can easily be evaluated and differentiated at any point. We show that due to the special reconstruction the WENO oscillation indicator can be computed in a mesh-independent manner by a simple quadratic functional. Our WENO scheme does not suffer from the problem of negative weights as previously described in the literature, since the linear weights are not used to increase accuracy. Accuracy is obtained by merely putting a large linear weight on the central stencil. The resulting one-step ADER finite volume scheme obtained in this way performs only one nonlinear WENO reconstruction per element and time step and thus can be implemented very efficiently even for unstructured grids in three space dimensions. We show convergence results obtained with the proposed method up to sixth order in space and time on unstructured triangular and tetrahedral grids in two and three space dimensions, respectively.

MSC:
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
76M12Finite volume methods (fluid mechanics)
35L45First order hyperbolic systems, initial value problems
WorldCat.org
Full Text: DOI
References:
[1] Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. Journal of computational physics 144, 45-58 (1994) · Zbl 0822.65062
[2] Atkins, H.; Shu, C. W.: Quadrature-free implementation of the discontinuous Galerkin method for hyperbolic equations. AIAA journal 36, 775-782 (1998)
[3] Balsara, D.; Shu, C. W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. Journal of computational physics 160, 405-452 (2000) · Zbl 0961.65078
[4] T.J. Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA paper no. 90-0013, 28th Aerospace Sciences Meeting January 1990.
[5] Ben-Artzi, M.; Falcovitz, J.: A second-order Godunov-type scheme for compressible fluid dynamics. Journal of computational physics 55, 1-32 (1984) · Zbl 0535.76070
[6] Butcher, J. C.: The numerical analysis of ordinary differential equations: Runge -- Kutta and general linear methods. (1987) · Zbl 0616.65072
[7] Casper, J.; Atkins, H. L.: A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems. Journal of computational physics 106, 62-76 (1993) · Zbl 0774.65066
[8] Cockburn, B.; Karniadakis, G. E.; Shu, C. W.: Discontinuous Galerkin methods. Lecture notes in computational science and engineering (2000)
[9] Davies-Jones, R.: Comments on ’a kinematic analysis of frontogenesis associated with a non-divergent vortex’. Journal of atmospheric sciences 42, 2073-2075 (1985)
[10] Dubiner, M.: Spectral methods on triangles and other domains. Journal of scientific computing 6, 345-390 (1991) · Zbl 0742.76059
[11] Dumbser, M.: Arbitrary high order schemes for the solution of hyperbolic conservation laws in complex domains. (2005)
[12] M. Dumbser, M. Käser, An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes II: The three-dimensional isotropic case, Geophysical Journal International, in press.
[13] Dumbser, M.; Munz, C. D.: ADER discontinuous Galerkin schemes for aeroacoustics. Comptes rendus mécanique 333, 683-687 (2005) · Zbl 1107.76044
[14] Dumbser, M.; Munz, C. D.: Arbitrary high order discontinuous Galerkin schemes. IRMA series in mathematics and theoretical physics, 295-333 (2005) · Zbl 1210.65165
[15] Dumbser, M.; Munz, C. D.: Building blocks for arbitrary high order discontinuous Galerkin schemes. Journal of scientific computing 27, 215-230 (2006) · Zbl 1115.65100
[16] Dumbser, M.; Schwartzkopff, T.; Munz, C. D.: Arbitrary high order finite volume schemes for linear wave propagation. Notes on numerical fluid mechanics and multidisciplinary design (NNFM), 129-144 (2006)
[17] Friedrich, O.: Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. Journal of computational physics 144, 194-212 (1998)
[18] Godunov, S. K.: Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics. Mathematics of the USSR-sbornik 47, 271-306 (1959)
[19] Harten, A.: High resolution schemes for hyperbolic conservation laws. Journal of computational physics 49, 357-393 (1983) · Zbl 0565.65050
[20] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R.: Uniformly high order accurate essentially non-oscillatory schemes III. Journal of computational physics 71, 231-303 (1987) · Zbl 0652.65067
[21] Hempel, D.: Local mesh adaption in two space dimensions. IMPACT computational science & engineering 5, 309-317 (1993) · Zbl 0795.65082
[22] D. Hempel, Isotropic refinement and recoarsening in 2 dimensions, Technical Report DLR IB 223-95 A 35, Deutsches Zentrum fnr Luft- und Raumfahrt (DLR), 1995.
[23] Hu, C.; Shu, C. W.: Weighted essentially non-oscillatory schemes on triangular meshes. Journal of computational physics 150, 97-127 (1999) · Zbl 0926.65090
[24] Iii, C. A. Doswell: A kinematic analysis of frontogenesis associated with a non-divergent vortex. Journal of atmospheric sciences 41, 1242-1248 (1984)
[25] Jiang, G. -S.; Shu, C. W.: Efficient implementation of weighted ENO schemes. Journal of computational physics, 202-228 (1996) · Zbl 0877.65065
[26] Käser, M.; Dumbser, M.: An arbitrary high order discontinuous Galerkin method for elastic waves on unstructured meshes I: The two-dimensional isotropic case with external source terms. Geophysical journal international 166, 855-877 (2006)
[27] Käser, M.; Iske, A.: ADER schemes on adaptive triangular meshes for scalar conservation laws. Journal of computational physics 205, 486-508 (2005) · Zbl 1072.65116
[28] Meister, A.; Struckmeier, J.: Hyperbolic partial differential equations. (2002) · Zbl 1012.65086
[29] Munz, C. D.: On the numerical dissipation of high resolution schemes for hyperbolic conservation laws. Journal of computational physics 77, 18-39 (1988) · Zbl 0646.65073
[30] Ollivier-Gooch, C.; Van Altena, M.: A high-order-accurate unstructured mesh finite-volume scheme for the advection -- diffusion equation. Journal of computational physics 181, 729-752 (2002) · Zbl 1178.76251
[31] Qiu, J.; Shu, C. W.: Hermite WENO schemes and their application as limiters for Runge -- Kutta discontinuous Galerkin method: one-dimensional case. Journal of computational physics 193, 115-135 (2003) · Zbl 1039.65068
[32] Qiu, J.; Shu, C. W.: Hermite WENO schemes and their application as limiters for Runge -- Kutta discontinuous Galerkin method II: Two dimensional case. Computers and fluids 34, 642-663 (2005) · Zbl 1134.65358
[33] Qiu, J.; Shu, C. W.: Runge -- Kutta discontinuous Galerkin method using WENO limiters. SIAM journal on scientific computing 26, 907-929 (2005) · Zbl 1077.65109
[34] Schwartzkopff, T.; Dumbser, M.; Munz, C. D.: Fast high order ADER schemes for linear hyperbolic equations. Journal of computational physics 197, 532-539 (2004) · Zbl 1052.65078
[35] Schwartzkopff, T.; Munz, C. D.; Toro, E. F.: ADER: a high order approach for linear hyperbolic systems in 2d. Journal of scientific computing 17, No. 1 -- 4, 231-240 (2002) · Zbl 1022.76034
[36] Shi, J.; Hu, C.; Shu, C. W.: A technique of treating negative weights in WENO schemes. Journal of computational physics 175, 108-127 (2002) · Zbl 0992.65094
[37] Sonar, T.: On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection. Computer methods in applied mechanics and engineering 140, 157-181 (1997) · Zbl 0898.76086
[38] Stroud, A. H.: Approximate calculation of multiple integrals. (1971) · Zbl 0379.65013
[39] Sweby, P. K.: High resolution TVD schemes using flux limiters. Lecture notes in applied mathematics 22, 289-309 (1985) · Zbl 0586.76119
[40] Titarev, V. A.; Toro, E. F.: ADER: arbitrary high order Godunov approach. Journal of scientific computing 17, No. 1 -- 4, 609-618 (2002) · Zbl 1024.76028
[41] Titarev, V. A.; Toro, E. F.: ADER schemes for three-dimensional nonlinear hyperbolic systems. Journal of computational physics 204, 715-736 (2005) · Zbl 1060.65641
[42] Toro, E. F.; Millington, R. C.; Nejad, L. A. M.: Towards very high order Godunov schemes. Godunov methods. Theory and applications, 905-938 (2001) · Zbl 0989.65094
[43] Toro, E. F.; Titarev, V.: TVD fluxes of the high-order ADER schemes. Journal of scientific computing 24, 285-309 (2005) · Zbl 1096.76029
[44] Toro, E. F.; Titarev, V. A.: Solution of the generalized Riemann problem for advection-reaction equations. Proceedings of royal society of London, 271-281 (2002) · Zbl 1019.35061
[45] Toro, E. F.; Titarev, V. A.: ADER schemes for scalar hyperbolic conservation laws with source terms in three space dimensions. Journal of computational physics 202, 196-215 (2005) · Zbl 1061.65103
[46] Van Leer, B.: Towards the ultimate conservative difference scheme II: Monotonicity and conservation combined in a second order scheme. Journal of computational physics 14, 361-370 (1974) · Zbl 0276.65055
[47] Van Leer, B.: Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’s method. Journal of computational physics 32, 101-136 (1979)
[48] Zalesak, S. T.: Fully multidimensional flux-corrected transport algorithms for fluids. Journal of computational physics 31, 335-362 (1979) · Zbl 0416.76002