##
**High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model.**
*(English)*
Zbl 1267.76010

Summary: We extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI time-integrators using the backward difference formulas from orders one to six. The reason for changing the time-integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high-order DG methods. Changing the time-integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element-type area integrals, but also the finite volume-type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations.

In this paper we couple the SI time-integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high-order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time-explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high-order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem.

The results show that the new high-order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high-order (HO) time-integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low-order time discretizations.

In this paper we couple the SI time-integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high-order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time-explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high-order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem.

The results show that the new high-order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high-order (HO) time-integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low-order time discretizations.

### MSC:

76B07 | Free-surface potential flows for incompressible inviscid fluids |

76M10 | Finite element methods applied to problems in fluid mechanics |

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

86-08 | Computational methods for problems pertaining to geophysics |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

### Keywords:

discontinuous Galerkin; explicit; finite element; finite volume; implicit; semi-implicit; shallow water; triangle### Software:

TsunAWI
PDF
BibTeX
XML
Cite

\textit{F. X. Giraldo} and \textit{M. Restelli}, Int. J. Numer. Methods Fluids 63, No. 9, 1077--1102 (2010; Zbl 1267.76010)

Full Text:
DOI

### References:

[1] | Alevras D. Simulations of the Indian Ocean tsunami with realistic bathymetry using a high-order triangular discontinuous Galerkin shallow water model. Masters Thesis, Naval Postgraduate School, 2008. |

[2] | Schwanenberg, Discontinuous Galerkin Methods pp 289– (2000) · Zbl 1041.76512 |

[3] | Li, The discontinuous Galerkin finite element method for the 2d shallow water equations, Mathematics and Computers in Simulation 56 pp 171– (2001) · Zbl 0987.76054 |

[4] | Aizinger, A discontinuous Galerkin method for two-dimensional flow and transport in shallow water, Advances in Water Resources 25 pp 67– (2002) · Zbl 1179.76049 |

[5] | Dupont, The adaptive spectral element method and comparisons with more traditional formulations for ocean modeling, Journal of Atmospheric and Oceanic Technology 21 pp 135– (2004) |

[6] | Eskilsson, A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations, International Journal for Numerical Methods in Fluids 45 pp 605– (2004) · Zbl 1085.76544 |

[7] | Remacle, An adaptive discretization of shallow-water equations based on discontinuous Galerkin methods, International Journal for Numerical Methods in Fluids 52 pp 903– (2006) · Zbl 1106.76044 |

[8] | Kubatko, HP discontinuous Galerkin methods for advection dominated problems in shallow water flow, Computer Methods in Applied Mechanics and Engineering 196 pp 437– (2006) |

[9] | Giraldo, A high-order triangular discontinuous Galerkin oceanic shallow water model, International Journal for Numerical Methods in Fluids 56 pp 899– (2008) · Zbl 1290.86002 |

[10] | Dolejsi, A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow, Journal of Computational Physics 198 pp 727– (2004) |

[11] | Dolejsi, Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems on nonconforming meshes, Computer Methods in Applied Mechanics and Engineering 196 pp 2813– (2007) |

[12] | Feistauer, On a robust discontinuous Galerkin technique for the solution of compressible flow, Journal of Computational Physics 224 pp 208– (2007) · Zbl 1114.76042 |

[13] | Feistauer, On the discontinuous Galerkin method for the simulation of compressible flow with wide range of Mach numbers, Computing and Visualization in Science 10 pp 17– (2007) |

[14] | Dolejsi, Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows, Communications in Computational Physics 4 pp 231– (2008) · Zbl 1364.76085 |

[15] | Restelli M. Semi-Lagrangian and semi-implicit discontinuous Galerkin methods for atmospheric modeling applications. Ph.D. Thesis, Politecnico di Milano, 2007. |

[16] | Restelli, A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling, SIAM Journal on Scientific Computing 31 pp 2231– (2009) · Zbl 1405.65127 |

[17] | Robert, An implicit time integration scheme for baroclinic models of the atmosphere, Monthly Weather Review 100 pp 329– (1972) |

[18] | Arcas, Sumatra tsunami: lessons from modeling, Surveys in Geophysics 27 pp 679– (2006) |

[19] | Geist, Implications of the 26 December 2004 Sumatra-Andaman earthquake on tsunami forecast and assessment models for great subduction zone earthquakes, Bulletin of the Seismological Society of America 97 pp 249– (2007) |

[20] | Kowalik, The tsunami of 26 December 2004: numerical modeling and energy considerations, Pure and Applied Geophysics 164 pp 379– (2007) |

[21] | Wang, Numerical simulations of the 2004 Indian Ocean tsunamis-coastal effects, Journal of Earthquake and Tsunami 1 pp 273– (2007) |

[22] | Harig, Tsunami simulations on several scales: comparison of approaches with unstructured meshes and nested grids, Ocean Dynamics 58 pp 429– (2008) |

[23] | Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM Journal on Numerical Analysis 35 pp 655– (1998) · Zbl 0933.41004 |

[24] | Taylor, An algorithm for computing Fekete points in the triangle, SIAM Journal on Numerical Analysis 38 pp 1707– (2000) · Zbl 0986.65017 |

[25] | Giraldo, High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere, Journal of Computational Physics 214 pp 447– (2006) · Zbl 1089.65096 |

[26] | Stroud, Approximate Calculation of Multiple Integrals (1971) · Zbl 0379.65013 |

[27] | Cools, Monomial cubature rules since Stroud: a compilation, Journal of Computational and Applied Mathematics 48 pp 309– (1993) · Zbl 0799.65027 |

[28] | Lyness, A survey of numerical cubature over triangles, Applied Mathematics 48 pp 127– (1994) · Zbl 0820.41026 |

[29] | Cools, Monomial cubature rules since Stroud: a compilation-part 2, Journal of Computational and Applied Mathematics 112 pp 21– (1999) · Zbl 0954.65021 |

[30] | Giraldo, Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations, Journal of Computational Physics 181 pp 499– (2002) · Zbl 1178.76268 |

[31] | Giraldo, A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in mesoscale nonhydrostatic atmospheric modeling: equation sets and test cases, Journal of Computational Physics 227 pp 3849– (2008) · Zbl 1194.76189 |

[32] | Cockburn, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing 16 pp 173– (2001) · Zbl 1065.76135 |

[33] | Spiteri, A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM Journal on Numerical Analysis 40 pp 469– (2002) · Zbl 1020.65064 |

[34] | De Luca TJ. Performance of hybrid Eulerian-Lagrangian semi-implicit time-integrators for nonhydrostatic mesoscale atmospheric modeling. Master’s Thesis, Naval Postgraduate School, 2007. |

[35] | Giraldo, A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations, Journal of Computational Physics 190 pp 623– (2003) · Zbl 1076.76058 |

[36] | Giraldo, Semi-implicit time-integrators for a scalable spectral element atmospheric model, Quarterly Journal of the Royal Meteorological Society 131 pp 2431– (2005) |

[37] | Giraldo, Hybrid Eulerian-Lagrangian semi-implicit time-integrators, Computers and Mathematics with Applications 52 pp 1325– (2006) · Zbl 1206.76038 |

[38] | Knoll, Jacobian-free Newton-Krylov methods: a survey of approaches and applications, Journal of Computational Physics 193 pp 357– (2004) |

[39] | Iskandarani, A staggered spectral element model with application to the oceanic shallow water equations, International Journal for Numerical Methods in Fluids 20 pp 393– (1995) · Zbl 0870.76057 |

[40] | Boyd, Equatorial solitary waves. Part 3: westward-travelling modons, Journal of Physical Oceanography 15 pp 46– (1985) |

[41] | Stommel, The westward intensification of wind-driven ocean currents, Transactions of the American Geophysics Union 29 pp 202– (1948) |

[42] | Toro, Shock-Capturing Methods for Free-Surface Shallow Flows pp 245– (2001) · Zbl 0996.76003 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.