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**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.

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:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

76M12 | Finite volume methods applied to problems in fluid mechanics |

35L45 | Initial value problems for first-order hyperbolic systems |

### Keywords:

WENO reconstruction; ADER approach; finite volume schemes; unstructured meshes; linear hyperbolic systems; numerical examples; discontinuous Galerkin finite element; convergence; weighted essential nonoscillatory (WENO); arbitrary high order schemes using derivatives (ADER)
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\textit{M. Dumbser} and \textit{M. Käser}, J. Comput. Phys. 221, No. 2, 693--723 (2007; Zbl 1110.65077)

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### 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), Wiley · 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), Springer |

[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), Shaker Verlag Aachen |

[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, (), 295-333 · 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, (), 129-144 |

[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) · Zbl 1392.76048 |

[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) · Zbl 0171.46204 |

[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] | Doswell, C.A., 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), Vieweg |

[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, 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), Prentice-Hall Inc. Englewood Cliffs, NJ · 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, 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, (), 905-938 · 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) · Zbl 1364.65223 |

[48] | Zalesak, S.T., Fully multidimensional flux-corrected transport algorithms for fluids, Journal of computational physics, 31, 335-362, (1979) · Zbl 0416.76002 |

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