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A stable dual pairing summation-by-parts method for sediment transport model with well-posed boundary conditions

Author: Prihandoko Rudi
Roberts Stephen
Duru Kenneth
Zoppou Christopher
Wiratama Kenny
Publication venue: Australian Mathematical Society
Publication date: 03/02/2026
Field of study

We develop a dual pairing summation-by-parts operator with Godunov flux splitting for the sediment transport model. The required number, location, and form of boundary conditions are determined via the energy method at the continuous level. Stability of the initial boundary value problem is rigorously established. At the discrete level the boundary conditions are weakly enforced via penalty terms. Stability of the numerical model is demonstrated through a discrete energy estimate that mimics the continuous energy. Numerical experiments are conducted to verify the analysis. References M. H. Carpenter, D. Gottlieb, and S. Abarbanel. Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111.2 (1994), pp. 220–236. doi: 10.1006/jcph.1994.1057 M. J. Castro Díaz, E. D. Fernández-Nieto, and A. M. Ferreiro. Sediment transport models in shallow water equations and numerical approach by high order finite volume methods. Comput. Fluids 37.3 (2008), pp. 299–316. doi: 10.1016/j.compfluid.2007.07.017 S. K. Godunov. A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sbornik 47.89 (1959). (English translation: I. Bohachevsky, Soviet Math. Dok., 11 (1969), pp. 494–497, https://hal.science/hal-01620642v1/), pp. 271–306. url: https://www.mathnet.ru/eng/sm4873 A. J. Grass. Sediment Transport by Waves and Currents. Science and Engineering Research Council (Great Britain), London Centre for Marine Technology. University College, London, Dept. of Civil Engineering. Report No. FL29. 1981. B. Gustafsson, H.-O. Kreiss, and J. Oliger. Time dependent problems and difference methods. John Wiley & Sons, 2013. doi: 10.1002/9781118548448 A. A. Kalinske. Criteria for determining sand-transport by surface-creep and saltation. Eos, Trans. Am. Geophys. Union 23 (2042), pp. 639–643. doi: 10.1029/TR023i002p00639 X. Liu and A. Beljadid. A coupled numerical model for water flow, sediment transport and bed erosion. Comput. Fluids 154 (2017), pp. 273–284. doi: 10.1016/j.compfluid.2017.06.013 T. Lundquist and J. Nordström. The SBP-SAT technique for time-discretization. 21st AIAA Computational Fluid Dynamics Conference. doi: 10.2514/6.2013-2834 T. Lundquist and J. Nordström. The SBP-SAT technique for initial value problems. J. Comput. Phys. 270 (2014), pp. 86–104. doi: 10.1016/j.jcp.2014.03.048 K. Mattsson. Diagonal-norm upwind SBP operators. J. Comput. Phys. 335 (2017), pp. 283–310. doi: 10.1016/j.jcp.2017.01.042 E. Meyer-Peter and R. Müller. Formulas for bed-load transport. TU Delft Repository. 1948. url: https://resolver.tudelft.nl/uuid:4fda9b61-be28-4703-ab06-43cdc2a21bd7/ P. Nielsen. Coastal Bottom Boundary Layers and Sediment Transport. Vol. 4. Advanced Series on Ocean Engineering. World Scientific, 1992. doi: 10.1142/1269 J. Nordström and M. H. Carpenter. High-Order finite difference methods, multidimensional linear problems, and curvilinear coordinates. J. Comput. Phys. 173.1 (2001), pp. 149–174. doi: 10.1006/jcph.2001.6864 R. Prihandoko, K. Duru, S. Roberts, and C. Zoppou. On well-posed boundary conditions and energy stable finite-volume method for the linear shallow water wave equation. ANZIAM J. 66.3 (2024), pp. 181–200. doi: 10.1017/S1446181124000191 L. C. van Rijn. Sediment transport, Part I: Bed load transport. J. Hyd. Eng. 110.10 (1984), pp. 1431–1456. doi: 10.1061/(ASCE)0733-9429(1984)110:10(1431) L. C. van Rijn. Sediment transport, Part II: Suspended load transport. J, Hyd. Eng. 110.11 (1984), pp. 1613–1641. doi: 10.1061/(ASCE)0733-9429(1984)110:11(1613) L. C. van Rijn. Sediment transport, Part III: Bed forms and alluvial roughness. J. Hyd. Eng. 110.12 (1984), pp. 1733–1754. doi: 10.1061/(ASCE)0733-9429(1984)110:12(1733) C. Williams and K. Duru. Full-spectrum dispersion relation preserving summation-by-parts operators. SIAM J. Numer. Anal. 62.4 (2024), pp. 1565–1588. doi: 10.1137/23M1586471 D. G. Wren, R. A. Kuhnle, E. J. Langendoen, and T. O. McAlpin. Sediment transport and bed topography for realistic unsteady flow hydrographs of varying length in a laboratory flume. J. Hyd. Eng. 150.4, 04024018 (2024). doi: 10.1061/JHEND8.HYENG-1376