Journal articles: 'Semi-Lagrangian discretizations'

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Bernard-Champmartin, Aude, Jean-Philippe Braeunig, Christophe Fochesato, and Thierry Goudon. "A Semi-Lagrangian Approach for Dilute Non-Collisional Fluid-Particle Flows." Communications in Computational Physics 19, no. 3 (March 2016): 801–40. http://dx.doi.org/10.4208/cicp.180315.110915a.

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AbstractWe develop numerical methods for the simulation of laden-flows where particles interact with the carrier fluid through drag forces. Semi-Lagrangian techniques are presented to handle the Vlasov-type equation which governs the evolution of the particles. We discuss several options to treat the coupling with the hydrodynamic system describing the fluid phase, paying attention to strategies based on staggered discretizations of the fluid velocity.

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Rivest, Chantal, Andrew Staniforth, and André Robert. "Spurious Resonant Response of Semi-Lagrangian Discretizations to Orographic Forcing: Diagnosis and Solution." Monthly Weather Review 122, no. 2 (February 1994): 366–76. http://dx.doi.org/10.1175/1520-0493(1994)122<0366:srrosl>2.0.co;2.

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Cordero, Elisabetta, and Andrew Staniforth. "A Problem with the Robert–Asselin Time Filter for Three-Time-Level Semi-Implicit Semi-Lagrangian Discretizations." Monthly Weather Review 132, no. 2 (February 2004): 600–610. http://dx.doi.org/10.1175/1520-0493(2004)132<0600:apwtrt>2.0.co;2.

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Guo, Wei, Ramachandran D. Nair, and Jing-Mei Qiu. "A Conservative Semi-Lagrangian Discontinuous Galerkin Scheme on the Cubed Sphere." Monthly Weather Review 142, no. 1 (January 1, 2014): 457–75. http://dx.doi.org/10.1175/mwr-d-13-00048.1.

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Abstract The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.

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Ardhuin, F., and T. H. C. Herbers. "Numerical and Physical Diffusion: Can Wave Prediction Models Resolve Directional Spread?" Journal of Atmospheric and Oceanic Technology 22, no. 7 (July 1, 2005): 886–95. http://dx.doi.org/10.1175/jtech1723.1.

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Abstract A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number N of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large N, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for N = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δt = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δt.

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Roy, Bruno, Pierre Poulin, and Eric Paquette. "Neural UpFlow." Proceedings of the ACM on Computer Graphics and Interactive Techniques 4, no. 3 (September 22, 2021): 1–26. http://dx.doi.org/10.1145/3480147.

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We present a novel up-resing technique for generating high-resolution liquids based on scene flow estimation using deep neural networks. Our approach infers and synthesizes small- and large-scale details solely from a low-resolution particle-based liquid simulation. The proposed network leverages neighborhood contributions to encode inherent liquid properties throughout convolutions. We also propose a particle-based approach to interpolate between liquids generated from varying simulation discretizations using a state-of-the-art bidirectional optical flow solver method for fluids in addition with a novel key-event topological alignment constraint. In conjunction with the neighborhood contributions, our loss formulation allows the inference model throughout epochs to reward important differences in regard to significant gaps in simulation discretizations. Even when applied in an untested simulation setup, our approach is able to generate plausible high-resolution details. Using this interpolation approach and the predicted displacements, our approach combines the input liquid properties with the predicted motion to infer semi-Lagrangian advection. We furthermore showcase how the proposed interpolation approach can facilitate generating large simulation datasets with a subset of initial condition parameters.

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Bonaventura, Luca, and Roberto Ferretti. "Flux form Semi-Lagrangian methods for parabolic problems." Communications in Applied and Industrial Mathematics 7, no. 3 (September 1, 2016): 56–73. http://dx.doi.org/10.1515/caim-2016-0022.

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Abstract A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and stability analysis is proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection diffusion and nonlinear parabolic problems.

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Filbet, Francis, and Charles Prouveur. "High order time discretization for backward semi-Lagrangian methods." Journal of Computational and Applied Mathematics 303 (September 2016): 171–88. http://dx.doi.org/10.1016/j.cam.2016.01.024.

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Yang, XueSheng, JiaBin Chen, JiangLin Hu, DeHui Chen, XueShun Shen, and HongLiang Zhang. "A semi-implicit semi-Lagrangian global nonhydrostatic model and the polar discretization scheme." Science in China Series D: Earth Sciences 50, no. 12 (December 2007): 1885–91. http://dx.doi.org/10.1007/s11430-007-0124-7.

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Shashkin, V. V., and M. A. Tolstykh. "Inherently mass-conservative version of the semi-Lagrangian absolute vorticity (SL-AV) atmospheric model dynamical core." Geoscientific Model Development 7, no. 1 (February 21, 2014): 407–17. http://dx.doi.org/10.5194/gmd-7-407-2014.

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Abstract. The semi-Lagrangian absolute vorticity (SL-AV) atmospheric model is the global semi-Lagrangian hydrostatic model used for operational medium-range and seasonal forecasts at the Hydrometeorological Centre of Russia. The distinct feature of the SL-AV dynamical core is the semi-implicit, semi-Lagrangian vorticity-divergence formulation on the unstaggered grid. A semi-implicit, semi-Lagrangian approach allows for long time steps but violates the global and local mass conservation. In particular, the total mass in simulations with semi-Lagrangian models can drift significantly if no a posteriori mass-fixing algorithm is applied. However, the global mass-fixing algorithms degrade the local mass conservation. The new inherently mass-conservative version of the SL-AV model dynamical core presented here ensures global and local mass conservation without mass-fixing algorithms. The mass conservation is achieved with the introduction of the finite-volume, semi-Lagrangian discretization for a continuity equation based on the 3-D extension of the conservative cascade semi-Lagrangian transport scheme (CCS). Numerical experiments show that the new version of the SL-AV dynamical core presented combines the accuracy and stability of the standard SL-AV dynamical core with the mass-conservation properties. The results of the mountain-induced Rossby-wave test and baroclinic instability test for the mass-conservative dynamical core are found to be in agreement with the results available in the literature.

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Shashkin, V. V., and M. A. Tolstykh. "Inherently mass-conservative version of the semi-Lagrangian Absolute Vorticity (SL-AV) atmospheric model dynamical core." Geoscientific Model Development Discussions 6, no. 3 (September 13, 2013): 4809–32. http://dx.doi.org/10.5194/gmdd-6-4809-2013.

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Abstract. The semi-Lagrangian Absolute Vorticity (SL-AV) atmospheric model is the global semi-Lagrangian hydrostatic model used for operational medium-range and seasonal forecasts at Hydrometeorological centre of Russia. The distinct feature of SL-AV dynamical core is the semi-implicit semi-Lagrangian vorticity-divergence formulation on the unstaggered grid. Semi-implicit semi-Lagrangian approach allows for long time steps while violates the global and local mass-conservation. In particular, the total mass in simulations with semi-Lagrangian models can drift significantly if no aposteriori mass-fixing algorithms are applied. However, the global mass-fixing algorithms degrade the local mass conservation. The inherently mass-conservative version of SL-AV model dynamical core presented in the article ensures global and local mass conservation without mass-fixing algorithms. The mass conservation is achieved with the introduction of the finite-volume semi-Lagrangian discretization for continuity equation based on the 3-D extension of the conservative cascade semi-Lagrangian transport scheme (CCS). The numerical experiments show that the presented new version of SL-AV dynamical core combines the accuracy and stability of the standard SL-AV dynamical core with the mass-conservation properties. The results of the mountain induced Rossby wave test and baroclinic instability test for mass-conservative dynamical core are found to be in agreement with the results available in literature.

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Staniforth, Andrew, Nigel Wood, and Sebastian Reich. "A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations." Quarterly Journal of the Royal Meteorological Society 132, no. 621C (October 2006): 3107–16. http://dx.doi.org/10.1256/qj.06.30.

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Payne, T. J. "A linear-stability analysis of the semi-implicit semi-Lagrangian discretization of the fully-compressible equations." Quarterly Journal of the Royal Meteorological Society 134, no. 632 (2008): 779–86. http://dx.doi.org/10.1002/qj.227.

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Wong, May, William C. Skamarock, Peter H. Lauritzen, Joseph B. Klemp, and Roland B. Stull. "Testing of a Cell-Integrated Semi-Lagrangian Semi-Implicit Nonhydrostatic Atmospheric Solver (CSLAM-NH) with Idealized Orography." Monthly Weather Review 143, no. 4 (March 31, 2015): 1382–98. http://dx.doi.org/10.1175/mwr-d-14-00059.1.

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Abstract A recently developed cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic atmospheric solver that uses the conservative semi-Lagrangian multitracer (CSLAM) transport scheme is extended to include orographic influences. With the introduction of a new semi-implicit CISL discretization of the continuity equation, the nonhydrostatic solver, called CSLAM-NH, has been shown to ensure inherently conservative and numerically consistent transport of air mass and other scalar variables, such as moisture and passive tracers. The extended CSLAM-NH presented here includes two main modifications: transformation of the equation set to a terrain-following height coordinate to incorporate orography and an iterative centered-implicit time-stepping scheme to enhance the stability of the scheme associated with gravity wave propagation at large time steps. CSLAM-NH is tested for a suite of idealized 2D flows, including linear mountain waves (dry), a downslope windstorm (dry), and orographic cloud formation.

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Subich, Christopher. "Instabilities in the Shallow-Water System with a Semi-Lagrangian, Time-Centered Discretization." Monthly Weather Review 150, no. 3 (March 2022): 467–80. http://dx.doi.org/10.1175/mwr-d-21-0054.1.

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Abstract Conventional wisdom suggests that the combination of semi-Lagrangian advection and an implicit treatment of gravity wave terms should result in a combined scheme for the shallow-water equations stable for high Courant numbers. This wisdom is well justified by linear analysis of the system about a uniform reference state with constant fluid depth and velocity, but it is only assumed to hold true in more complex scenarios. This work finds that this conventional wisdom no longer holds in more complicated flow regimes, in particular when the background state is given by steady-state flow past topography. Instead, this background state admits a wide range of instabilities that can lead to noise in atmospheric forecasts. Significance Statement This work shows that solutions to the shallow-water equations with a semi-Lagrangian treatment of advection and an implicit, time-centered treatment of gravity wave terms can be unstable when there is a background state of flow over topography. This basic algorithm is used by many operational weather-forecasting models to simulate the meteorological equations, and showing an instability in the simplified, shallow-water system suggests that a similar mechanism may be responsible for “noise” in operational weather forecasts under some circumstances. If this problem can be addressed, it could allow numerical weather models to operate with less dissipation, improving forecast quality.

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Wensch, Jörg, and Andreas Naumann. "Semi-Lagrangian discretization of the upper-convective derivative in Non-Newtonian fluid flow." PAMM 12, no. 1 (December 2012): 765–66. http://dx.doi.org/10.1002/pamm.201210371.

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Wood, Nigel, Andrew Staniforth, Andy White, Thomas Allen, Michail Diamantakis, Markus Gross, Thomas Melvin, et al. "An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations." Quarterly Journal of the Royal Meteorological Society 140, no. 682 (December 4, 2013): 1505–20. http://dx.doi.org/10.1002/qj.2235.

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Meneses, Lisiane Ramires, and Ricardo Carvalho Almeida. "MODELO SEMI-LAGRANGEANO DE DISPERSÃO ATMOSFÉRICA - AVALIAÇÃO." Ciência e Natura 38 (July 20, 2016): 418. http://dx.doi.org/10.5902/2179460x20307.

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A two-dimensional Semi-Lagrangian numerical model to simulate pollutant dispersion in the atmospheric boundary layer on flat terrain is tested. The model is based on the numerical solution of the advection-diffusion equation using a three-time-level Semi-Lagrangian scheme for the discretization of the advection term. The problem of pollutant dispersion in stable and unstable atmospheric conditions is investigated using two different parameterizations for the vertical turbulent diffusion coefficient. The performance of the model is discussed from the confrontation of the concentration values observed in the dispersion experiment of Prairie Grass with the ones simulated by the model. The statistical analysis shows that the model simulates the experimental data in a satisfactory way and presents results similar to the ones obtained by other authors using the models available in the literature.

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Tolstykh, Mikhail, Vladimir Shashkin, Rostislav Fadeev, and Gordey Goyman. "Vorticity-divergence semi-Lagrangian global atmospheric model SL-AV20: dynamical core." Geoscientific Model Development 10, no. 5 (May 22, 2017): 1961–83. http://dx.doi.org/10.5194/gmd-10-1961-2017.

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Abstract. SL-AV (semi-Lagrangian, based on the absolute vorticity equation) is a global hydrostatic atmospheric model. Its latest version, SL-AV20, provides global operational medium-range weather forecast with 20 km resolution over Russia. The lower-resolution configurations of SL-AV20 are being tested for seasonal prediction and climate modeling. The article presents the model dynamical core. Its main features are a vorticity-divergence formulation at the unstaggered grid, high-order finite-difference approximations, semi-Lagrangian semi-implicit discretization and the reduced latitude–longitude grid with variable resolution in latitude. The accuracy of SL-AV20 numerical solutions using a reduced lat–lon grid and the variable resolution in latitude is tested with two idealized test cases. Accuracy and stability of SL-AV20 in the presence of the orography forcing are tested using the mountain-induced Rossby wave test case. The results of all three tests are in good agreement with other published model solutions. It is shown that the use of the reduced grid does not significantly affect the accuracy up to the 25 % reduction in the number of grid points with respect to the regular grid. Variable resolution in latitude allows us to improve the accuracy of a solution in the region of interest.

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Al-Mosallam, Mohammed, and Kensuke Yokoi. "Efficient Implementation of Volume/Surface Integrated Average-Based Multi-Moment Method." International Journal of Computational Methods 14, no. 02 (February 22, 2017): 1750010. http://dx.doi.org/10.1142/s0219876217500104.

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We investigated discretization strategies of the conservation equation in volume/surface integrated average-based multi-moment method (VSIAM3) which is a numerical framework for incompressible and compressible flows based on a multi-moment concept. We investigated these strategies through the lid-driven cavity flow problem, shock tube problems, 2D explosion test and droplet splashing on a superhydrophobic substrate. We found that the use of the constrained interpolation profile-conservative semi-Lagrangian with rational function (CIP-CSLR) method as the conservation equation solver is critically important for the robustness of incompressible flow simulations using VSIAM3 and that numerical results are sensitive to discretization techniques of the divergence term in the conservation equation. Based on these results, we proposed efficient implementation techniques of VSIAM3.

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Diamantakis, Michail, and Linus Magnusson. "Sensitivity of the ECMWF Model to Semi-Lagrangian Departure Point Iterations." Monthly Weather Review 144, no. 9 (September 2016): 3233–50. http://dx.doi.org/10.1175/mwr-d-15-0432.1.

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Accurate estimation of the position of the departure points (d.p.) is crucial for the accuracy of a semi-Lagrangian NWP model. This calculation is often performed applying an implicit discretization to a kinematic equation solved by a fixed-point iteration scheme. A small number of iterations is typically used, assuming that this is sufficient for convergence. This assumption, derived from a past theoretical analysis, is revisited here. Analyzing the convergence of a generic d.p. iteration scheme and testing the ECMWF Integrated Forecast System (IFS) model, it is demonstrated that 2–3 iterations may not be sufficient for convergence to satisfactory accuracy in a modern high-resolution global model. Large forecast improvements can be seen by increasing the number of iterations. The extratropical geopotential error decreases and the simulated vertical structure of tropical cyclones improves. These findings prompted the implementation of an algorithm in which stopping criteria based on estimated convergence rates are used to “dynamically” stop d.p. iterations when an error tolerance criterion is satisfied. This is applied consistently to the nonlinear forecast, tangent linear, and adjoint models used by the ECMWF data assimilation system (4DVAR). Although the additional benefit of dynamic iteration is small, its testing reinforces the conclusion that a larger number of iterations is needed in regions of strong winds and shear. Furthermore, experiments suggest that dynamic iteration may prevent occasional 4DVAR failures in cases of strong stratospheric cross-polar flow in which the tangent linear model becomes unstable.

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Neilan, Michael, Abner J. Salgado, and Wujun Zhang. "Numerical analysis of strongly nonlinear PDEs." Acta Numerica 26 (May 1, 2017): 137–303. http://dx.doi.org/10.1017/s0962492917000071.

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We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

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Natarajan, Sundararajan, and Chandramouli Padmanabhan. "Scaled Boundary Finite Element Method for Mid-Frequency Acoustics of Cavities." Journal of Theoretical and Computational Acoustics 29, no. 01 (March 2021): 2150001. http://dx.doi.org/10.1142/s2591728521500018.

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In this paper, a semi-analytical framework, based on the scaled boundary finite element method (SBFEM), is proposed, to study interior acoustic problems in the mid-frequency range in two and three dimensions. The SBFEM shares the advantages of both the finite element method (FEM) and the boundary element method (BEM). Like the FEM, it does not require the fundamental solution (Green’s function) and similar to the BEM only the boundary is discretized, thus reducing the spatial dimensionality by one. The solution within the domain is represented analytically, while on the boundary, it is represented by finite elements. Although different boundary representations are possible, only the Lagrangian description is investigated in this paper. The proposed framework is validated using closed-form solutions, and direct comparisons are made with the conventional FEM based on Lagrangian description; this will be demonstrated using two 2D cavities from literature as well as one 3D cavity. The improved accuracy and reduced computational time can be attributed to the semi-analytical formulation combined with the boundary discretization.

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Carcano, S., L. Bonaventura, T. Esposti Ongaro, and A. Neri. "A semi-implicit, second-order-accurate numerical model for multiphase underexpanded volcanic jets." Geoscientific Model Development 6, no. 6 (November 4, 2013): 1905–24. http://dx.doi.org/10.5194/gmd-6-1905-2013.

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Abstract. An improved version of the PDAC (Pyroclastic Dispersal Analysis Code, Esposti Ongaro et al., 2007) numerical model for the simulation of multiphase volcanic flows is presented and validated for the simulation of multiphase volcanic jets in supersonic regimes. The present version of PDAC includes second-order time- and space discretizations and fully multidimensional advection discretizations in order to reduce numerical diffusion and enhance the accuracy of the original model. The model is tested on the problem of jet decompression in both two and three dimensions. For homogeneous jets, numerical results are consistent with experimental results at the laboratory scale (Lewis and Carlson, 1964). For nonequilibrium gas–particle jets, we consider monodisperse and bidisperse mixtures, and we quantify nonequilibrium effects in terms of the ratio between the particle relaxation time and a characteristic jet timescale. For coarse particles and low particle load, numerical simulations well reproduce laboratory experiments and numerical simulations carried out with an Eulerian–Lagrangian model (Sommerfeld, 1993). At the volcanic scale, we consider steady-state conditions associated with the development of Vulcanian and sub-Plinian eruptions. For the finest particles produced in these regimes, we demonstrate that the solid phase is in mechanical and thermal equilibrium with the gas phase and that the jet decompression structure is well described by a pseudogas model (Ogden et al., 2008). Coarse particles, on the other hand, display significant nonequilibrium effects, which associated with their larger relaxation time. Deviations from the equilibrium regime, with maximum velocity and temperature differences on the order of 150 m s−1 and 80 K across shock waves, occur especially during the rapid acceleration phases, and are able to modify substantially the jet dynamics with respect to the homogeneous case.

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Carcano, S., L. Bonaventura, T. Esposti Ongaro, and A. Neri. "A semi-implicit, second order accurate numerical model for multiphase underexpanded volcanic jets." Geoscientific Model Development Discussions 6, no. 1 (January 22, 2013): 399–452. http://dx.doi.org/10.5194/gmdd-6-399-2013.

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Abstract. An improved version of the PDAC (Pyroclastic Dispersal Analysis Code, Esposti Ongaro et al., 2007) numerical model for the simulation of multiphase volcanic flows is presented and validated for the simulation of multiphase volcanic jets in supersonic regimes. The present version of PDAC includes second-order time and space discretizations and fully multidimensional advection discretizations, in order to reduce numerical diffusion and enhance the accuracy of the original model. The model is tested on the problem of jet decompression, in both two and three dimensions. For homogeneous jets, numerical results are consistent with experimental results at the laboratory scale (Lewis and Carlson, 1964). For non-equilibrium gas-particle jets, we consider monodisperse and bidisperse mixtures and we quantify non-equilibrium effects in terms of the ratio between the particle relaxation time and a characteristic jet time scale. For coarse particles and low particle load, numerical simulations well reproduce laboratory experiments and numerical simulations carried out with an Eulerian-Lagrangian model (Sommerfeld, 1993). At the volcanic scale, we consider steady-state conditions associated to the development of Vulcanian and sub-Plinian eruptions. For the finest particles produced in these regimes, we demonstrate that the solid phase is in mechanical and thermal equilibrium with the gas phase and that the jet decompression structure is well described by a pseudogas model (Ogden et al., 2008). Coarse particles, on the contrary, display significant non-equilibrium effects, associated to their larger relaxation time. Deviations from the equilibrium regime occur especially during the rapid acceleration phases and are able to appreciably modify the average jet dynamics, with maximum velocity and temperature differences of the order of 150 m s−1 and 80 K across shock waves.

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Nikitin, Kirill D., Maxim A. Olshanskii, Kirill M. Terekhov, and Yuri V. Vassilevski. "A splitting method for free surface flows over partially submerged obstacles." Russian Journal of Numerical Analysis and Mathematical Modelling 33, no. 2 (April 25, 2018): 95–110. http://dx.doi.org/10.1515/rnam-2018-0009.

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Abstract The paper proposes a stable time-splitting method for the numerical simulation of free-surface viscous flows. The key features of the method are a semi-Lagrangian scheme for the level-set function transport improved with MacCormack predictor–corrector step with limiting strategy and an adaptive volume-correction procedure. The spatial discretization is done by a hybrid finite volume/finite difference method on dynamically adaptive hexahedral meshes. Numerical verification is done by comparing full-scale 3D numerical simulations of the sloshing tank and the coastal wave run-up with other numerical and experimental results known from the literature.

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Asmouh, Ilham, Mofdi El-Amrani, Mohammed Seaid, and Naji Yebari. "A Cell-Centered Semi-Lagrangian Finite Volume Method for Solving Two-Dimensional Coupled Burgers’ Equations." Computational and Mathematical Methods 2022 (February 13, 2022): 1–18. http://dx.doi.org/10.1155/2022/8192192.

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A cell-centered finite volume semi-Lagrangian method is presented for the numerical solution of two-dimensional coupled Burgers’ problems on unstructured triangular meshes. The method combines a modified method of characteristics for the time integration and a cell-centered finite volume for the space discretization. The new method belongs to fractional-step algorithms for which the convection and the viscous parts in the coupled Burgers’ problems are treated separately. The crucial step of interpolation in the convection step is performed using two local procedures accounting for the element where the departure point is located. The resulting semidiscretized system is then solved using a third-order explicit Runge-Kutta scheme. In contrast to the Eulerian-based methods, we apply the new method for each time step along the characteristic curves instead of the time direction. The performance of the current method is verified using different examples for coupled Burgers’ problems with known analytical solutions. We also apply the method for simulation of an example of coupled Burgers’ flows in a complex geometry. In these test problems, the new cell-centered finite volume semi-Lagrangian method demonstrates its ability to accurately resolve the two-dimensional coupled Burgers’ problems.

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Tumolo, Giovanni. "A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals." Communications in Applied and Industrial Mathematics 7, no. 3 (September 1, 2016): 165–90. http://dx.doi.org/10.1515/caim-2016-0026.

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Abstract As an extension of a previous work considering a fully advective formulation on Cartesian meshes, a mass conservative discretization approach is presented here for the shallow water equations, based on discontinuous finite elements on general structured meshes of quadrilaterals. A semi-implicit time integration is performed by employing the TR-BDF2 scheme and is combined with the semi-Lagrangian technique for the momentum equation only. Indeed, in order to simplify the derivation of the discrete linear Helmoltz equation to be solved at each time-step, a non-conservative formulation of the momentum equation is employed. The Eulerian flux form is considered instead for the continuity equation in order to ensure mass conservation. Numerical results show that on distorted meshes and for relatively high polynomial degrees, the proposed numerical method fully conserves mass and presents a higher level of accuracy than a standard off-centered Crank Nicolson approach. This is achieved without any significant imprinting of the mesh distortion on the solution.

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Bénard, P., and K. Yessad. "On the Discretization of the Continuity Equation in Semi-Lagrangian Models with a Pressure-Type Vertical Coordinate." Monthly Weather Review 129, no. 7 (July 2001): 1746–49. http://dx.doi.org/10.1175/1520-0493(2001)129<1746:otdotc>2.0.co;2.

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Bonaventura, Luca, Roberto Ferretti, and Lorenzo Rocchi. "A fully semi-Lagrangian discretization for the 2D incompressible Navier–Stokes equations in the vorticity-streamfunction formulation." Applied Mathematics and Computation 323 (April 2018): 132–44. http://dx.doi.org/10.1016/j.amc.2017.11.030.

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Li, Xingliang, Dehui Chen, Xindong Peng, Keiko Takahashi, and Feng Xiao. "A Multimoment Finite-Volume Shallow-Water Model on the Yin–Yang Overset Spherical Grid." Monthly Weather Review 136, no. 8 (August 1, 2008): 3066–86. http://dx.doi.org/10.1175/2007mwr2206.1.

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Abstract A numerical model for shallow-water equations has been built and tested on the Yin–Yang overset spherical grid. A high-order multimoment finite-volume method is used for the spatial discretization in which two kinds of so-called moments of the physical field [i.e., the volume integrated average (VIA) and the point value (PV)] are treated as the model variables and updated separately in time. In the present model, the PV is computed by the semi-implicit semi-Lagrangian formulation, whereas the VIA is predicted in time via a flux-based finite-volume method and is numerically conserved on each component grid. The concept of including an extra moment (i.e., the volume-integrated value) to enforce the numerical conservativeness provides a general methodology and applies to the existing semi-implicit semi-Lagrangian formulations. Based on both VIA and PV, the high-order interpolation reconstruction can only be done over a single grid cell, which then minimizes the overlapping zone between the Yin and Yang components and effectively reduces the numerical errors introduced in the interpolation required to communicate the data between the two components. The present model completely gets around the singularity and grid convergence in the polar regions of the conventional longitude–latitude grid. Being an issue demanding further investigation, the high-order interpolation across the overlapping region of the Yin–Yang grid in the current model does not rigorously guarantee the numerical conservativeness. Nevertheless, these numerical tests show that the global conservation error in the present model is negligibly small. The model has competitive accuracy and efficiency.

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Untch, A., and M. Hortal. "A finite-element scheme for the vertical discretization of the semi-Lagrangian version of the ECMWF forecast model." Quarterly Journal of the Royal Meteorological Society 130, no. 599 (April 2004): 1505–30. http://dx.doi.org/10.1256/qj.03.173.

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Kamensky, David, John A. Evans, and Ming-Chen Hsu. "Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis." Communications in Computational Physics 18, no. 4 (October 2015): 1147–80. http://dx.doi.org/10.4208/cicp.150115.170415s.

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AbstractThe purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, T.J.R. Hughes, “An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves”, Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.

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Kühnlein, Christian, Willem Deconinck, Rupert Klein, Sylvie Malardel, Zbigniew P. Piotrowski, Piotr K. Smolarkiewicz, Joanna Szmelter, and Nils P. Wedi. "FVM 1.0: a nonhydrostatic finite-volume dynamical core for the IFS." Geoscientific Model Development 12, no. 2 (February 13, 2019): 651–76. http://dx.doi.org/10.5194/gmd-12-651-2019.

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Abstract. We present a nonhydrostatic finite-volume global atmospheric model formulation for numerical weather prediction with the Integrated Forecasting System (IFS) at ECMWF and compare it to the established operational spectral-transform formulation. The novel Finite-Volume Module of the IFS (henceforth IFS-FVM) integrates the fully compressible equations using semi-implicit time stepping and non-oscillatory forward-in-time (NFT) Eulerian advection, whereas the spectral-transform IFS solves the hydrostatic primitive equations (optionally the fully compressible equations) using a semi-implicit semi-Lagrangian scheme. The IFS-FVM complements the spectral-transform counterpart by means of the finite-volume discretization with a local low-volume communication footprint, fully conservative and monotone advective transport, all-scale deep-atmosphere fully compressible equations in a generalized height-based vertical coordinate, and flexible horizontal meshes. Nevertheless, both the finite-volume and spectral-transform formulations can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude–latitude coordinates, and physics parameterizations, thereby facilitating their comparison, coexistence, and combination in the IFS. We highlight the advanced semi-implicit NFT finite-volume integration of the fully compressible equations of IFS-FVM considering comprehensive moist-precipitating dynamics with coupling to the IFS cloud parameterization by means of a generic interface. These developments – including a new horizontal–vertical split NFT MPDATA advective transport scheme, variable time stepping, effective preconditioning of the elliptic Helmholtz solver in the semi-implicit scheme, and a computationally efficient implementation of the median-dual finite-volume approach – provide a basis for the efficacy of IFS-FVM and its application in global numerical weather prediction. Here, numerical experiments focus on relevant dry and moist-precipitating baroclinic instability at various resolutions. We show that the presented semi-implicit NFT finite-volume integration scheme on co-located meshes of IFS-FVM can provide highly competitive solution quality and computational performance to the proven semi-implicit semi-Lagrangian integration scheme of the spectral-transform IFS.

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Vázquez-Quesada, Adolfo, and Marco Ellero. "GENERIC-compliant simulations of Brownian multi-particle systems: modeling stochastic lubrication." SeMA Journal 79, no. 1 (January 8, 2022): 165–85. http://dx.doi.org/10.1007/s40324-021-00280-z.

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AbstractA stochastic Lagrangian model for simulating the dynamics and rheology of a Brownian multi-particle system interacting with a simple liquid medium is presented. The discrete particle model is formulated within the GENERIC framework for Non-Equilibrium Thermodynamics and therefore it satisfies discretely the First/Second Laws of Thermodynamics and the Fluctuation Dissipation Theorem (FDT). Long-range fluctuating hydrodynamics interactions between suspended particles are described by an explicit solvent model. To this purpose, the Smoothed Dissipative Particle Dynamics method is adopted, which is a GENERIC-compliant Lagrangian meshless discretization of the fluctuating Navier–Stokes equations. In dense multi-particle systems, the average inter-particle distance is typically small compared to the particle size and short-range hydrodynamics interactions play a major role. In order to bypass an explicit—computationally costly—solution for these forces, a lubrication correction is introduced based on semi-analytical expressions for spheres under Stokes flow conditions. We generalize here the lubrication formalism to Brownian conditions, where an additional thermal-lubrication contribution needs to be taken into account in a way that discretely satisfies FDT. The coupled lubrication dynamics is integrated in time using a generalized semi-implicit splitting scheme for stochastic differential equations. The model is finally validated for a single particle diffusion as well as for a Brownian multi-particle system under homogeneous shear flow. Results for the diffusional properties as well as the rheological behavior of the whole suspension are presented and discussed.

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Thuburn, J., M. Zerroukat, N. Wood, and A. Staniforth. "Coupling a mass-conserving semi-Lagrangian scheme (SLICE) to a semi-implicit discretization of the shallow-water equations: Minimizing the dependence on a reference atmosphere." Quarterly Journal of the Royal Meteorological Society 136, no. 646 (January 2010): 146–54. http://dx.doi.org/10.1002/qj.517.

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Staniforth, Andrew, and Nigel Wood. "Analysis of the response to orographic forcing of a time-staggered semi-Lagrangian discretization of the rotating shallow-water equations." Quarterly Journal of the Royal Meteorological Society 132, no. 621C (October 2006): 3117–26. http://dx.doi.org/10.1256/qj.06.51.

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Girard, Claude, André Plante, Michel Desgagné, Ron McTaggart-Cowan, Jean Côté, Martin Charron, Sylvie Gravel, et al. "Staggered Vertical Discretization of the Canadian Environmental Multiscale (GEM) Model Using a Coordinate of the Log-Hydrostatic-Pressure Type." Monthly Weather Review 142, no. 3 (March 1, 2014): 1183–96. http://dx.doi.org/10.1175/mwr-d-13-00255.1.

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Abstract The Global Environmental Multiscale (GEM) model is the Canadian atmospheric model used for meteorological forecasting at all scales. A limited-area version now also exists. It is a gridpoint model with an implicit semi-Lagrangian iterative space–time integration scheme. In the “horizontal,” the equations are written in spherical coordinates with the traditional shallow atmosphere approximations and are discretized on an Arakawa C grid. In the “vertical,” the equations were originally defined using a hydrostatic-pressure coordinate and discretized on a regular (unstaggered) grid, a configuration found to be particularly susceptible to noise. Among the possible alternatives, the Charney–Phillips grid, with its unique characteristics, and, as the vertical coordinate, log-hydrostatic pressure are adopted. In this paper, an attempt is made to justify these two choices on theoretical grounds. The resulting equations and their vertical discretization are described and the solution method of what is forming the new dynamical core of GEM is presented, focusing on these two aspects.

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Besse, Nicolas, Norbert Mauser, and Eric Sonnendrücker. "Numerical Approximation of Self-Consistent Vlasov Models for Low-Frequency Electromagnetic Phenomena." International Journal of Applied Mathematics and Computer Science 17, no. 3 (October 1, 2007): 361–74. http://dx.doi.org/10.2478/v10006-007-0030-3.

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Numerical Approximation of Self-Consistent Vlasov Models for Low-Frequency Electromagnetic PhenomenaWe present a new numerical method to solve the Vlasov-Darwin and Vlasov-Poisswell systems which are approximations of the Vlasov-Maxwell equation in the asymptotic limit of the infinite speed of light. These systems model low-frequency electromagnetic phenomena in plasmas, and thus "light waves" are somewhat supressed, which in turn allows the numerical discretization to dispense with the Courant-Friedrichs-Lewy condition on the time step. We construct a numerical scheme based on semi-Lagrangian methods and time splitting techniques. We develop a four-dimensional phase space algorithm for the distribution function while the electromagnetic field is solved on a two-dimensional Cartesian grid. Finally, we present two nontrivial test cases: (a) the wave Landau damping and (b) the electromagnetic beam-plasma instability. For these cases our numerical scheme works very well and is in agreement with analytic kinetic theory.

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Vanella, Marcos, Kevin McGrattan, Randall McDermott, Glenn Forney, William Mell, Emanuele Gissi, and Paolo Fiorucci. "A Multi-Fidelity Framework for Wildland Fire Behavior Simulations over Complex Terrain." Atmosphere 12, no. 2 (February 18, 2021): 273. http://dx.doi.org/10.3390/atmos12020273.

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A method for the large-eddy simulation (LES) of wildfire spread over complex terrain is presented. In this scheme, a cut-cell immersed boundary method (CC-IBM) is used to render the complex terrain, defined by a tessellation, on a rectilinear Cartesian grid. Discretization of scalar transport equations for chemical species is done via a finite volume scheme on cut-cells defined by the intersection of the terrain geometry and the Cartesian cells. Momentum transport and heat transfer close to the immersed terrain are handled using dynamic wall models and a direct forcing immersed boundary method. A new “open” convective inflow/outflow method for specifying atmospheric wind boundary conditions is presented. Additionally, three basic approaches have been explored to model fire spread: (1) Representing the vegetation as a collection of Lagrangian particles, (2) representing the vegetation as a semi-porous boundary, and (3) representing the fire spread using a level set method, in which the fire spreads as a function of terrain slope, vegetation type, and wind speed. Several test and validation cases are reported to demonstrate the capabilities of this novel wildfire simulation methodology.

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Hante, Stefan, Denise Tumiotto, and Martin Arnold. "A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model." Multibody System Dynamics 54, no. 1 (December 6, 2021): 97–123. http://dx.doi.org/10.1007/s11044-021-09807-8.

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AbstractIn this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

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Lu, Chengwei, Jianzhong Zhou, Dechao Hu, Shuai Yuan, and Yi Liu. "Fast and high-precision simulation of hydrodynamic and water quality process in river networks." MATEC Web of Conferences 246 (2018): 01073. http://dx.doi.org/10.1051/matecconf/201824601073.

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To realize a fast and high-precision simulation of unsteady flows and water quality process in river networks, a 1-D mathematical modelling system for free-surface flows and water quality process is developed. In the hydrodynamic model, a θ semi-implicit method is used to discretize the free-surface gradient term in the momentum equation while the Eulerian-Lagrangian method is employed to solve the advection term. To achieve good mass conservation, the finite-volume method is used to discretize continuity equation. Moreover, the resulting hydrodynamic model is unconditionally stable with respect to the gravity wave speed and flow advection. In solving the scalar transport equation, the forward difference is used for discretization in time while the centre-difference is in space. A sub-cycling is introduced to divide the computational time step and improve the stability of water quality model. Therefore, large time steps are allowed in both the hydrodynamic and the transport models. The coupling of the branches of the river networks is solved using a predictioncorrection method. Then, a Gaussian pulse test and a rectangular wave test are employed to demonstrate the accuracy and the efficiency of the system, which will also provide powerful supports for ecological operations of cascade reservoirs in drainage basins.

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Liu, Cheng, Ruoqing Gao, and Changhong Hu. "An approximated volume of fluid method with the modified height function method in the simulation of surface tension driven flows." AIP Advances 12, no. 8 (August 1, 2022): 085308. http://dx.doi.org/10.1063/5.0098717.

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Surface tension in two-phase flow problems plays a dominant role in many micro-flow phenomena and has an important influence on the development of flow instability phenomena that contain free surfaces. In this study, the multi-moment finite volume method is extended for direct numerical simulation of two-phase flow problems. A constraint interpolation profile–CSL (semi-Lagrangian) scheme is used for discretization of the advection part in the momentum equation. A compact volume of fluid method–approximated piecewise linear calculation method without flux limiter is proposed for capturing the moving interface. For modeling the surface tension accurately, the logic in curvature estimation is redesigned based on the height function (HF) method. The isolated volumetric fractions that may reduce accuracy in HF integration are excluded, and the numerical solution shows that the accuracy in the curvature estimation is improved for a coarse mesh. The present method is implemented with a parallel block-structured adaptive mesh refinement (BAMR) strategy; thus, the computational cost can be reduced significantly. Numerical tests show that the present BAMR solver is capable of reproducing the theoretical predictions of capillary wave instability problems with high accuracy. The simulation of droplet collisions further demonstrates the accuracy of the surface tension model. Finally, we extend it to the liquid jet atomization. The wavy disturbance, film breakup, liquid filament pinch-off, and droplet generation are well reproduced. The droplet size distribution satisfies the experimental measurement and theoretical predictions power-law. BAMR shows a huge advantage in computational efficiency than the traditional Cartesian grid. The findings of this study can help for a better understanding of the micro-mechanism of surface tension driven flows.

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Mola, Andrea, Luca Heltai, and Antonio DeSimone. "Wet and Dry Transom Stern Treatment for Unsteady and Nonlinear Potential Flow Model for Naval Hydrodynamics Simulations." Journal of Ship Research 61, no. 01 (March 1, 2017): 1–14. http://dx.doi.org/10.5957/jsr.2017.61.1.1.

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We present a model for the fast evaluation of the total drag of ship hulls operating in both wet and dry transom stern conditions, in calm or wavy water, based on the combination of an unsteady semi-Lagrangian potential flow formulation with fully nonlinear free-surface treatment, experimental correlations, and simplified viscous drag modeling. The implementation is entirely based on open source libraries. The spatial discretization is solved using a streamline upwind Petrov-Galerkin stabilization of an iso-parametric, collocation based, boundary element method, implemented using the open source library deal. II. The resulting nonlinear differential-algebraic system is integrated in time using implicit backward differentiation formulas, implemented in the open source library SUNDIALS. The Open CASCADE library is used to interface the model directly with computer-aided design data structures. The model accounts automatically for hulls with a transom stern, both in wet and dry regimes, by using a specific treatment of the free-surface nodes on the stern edge that automatically detects when the hull advances at low speeds. In this case, the transom stern is partially immersed, and a pressure patch is applied on the water surface detaching from the transom stern, to recover the gravity effect of the recirculating water on the underlying irrotational flow domain. The parameters of the model used to impose the pressure patch are approximated from experimental relations found in the literature. The test cases considered are those of the U.S. Navy Combatant DTMB-5415 and the National Physical Laboratory hull. Comparisons with experimental data on quasi-steady test cases for both water elevation and total hull drag are presented and discussed. The quality of the results obtained on quasi-steady simulations suggests that this model can represent a promising alternative to current unsteady solvers for simulations with Froude numbers below 0.35.

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Staniforth, Andrew, and Nigel Wood. "Comments on 'A finite-element scheme for the vertical discretization in the semi-Lagrangian version of the ECMWF forecast model' by A. Untch and M. Hortal (April B, 2004, 130, 1505–1530)." Quarterly Journal of the Royal Meteorological Society 131, no. 606 (January 1, 2005): 765–72. http://dx.doi.org/10.1256/qj.04.10.

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Sen, Prakash Kumar, Mahesh Bhiwapurkar, and S. P. Harsha. "A Numerical Simulation of UIC60 Rail-Weld's Fatigue and Crack Growth under Wheel Frictional Contact and Bending." Advanced Engineering Forum 45 (April 4, 2022): 31–42. http://dx.doi.org/10.4028/p-27z21t.

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In service condition rail joints, especially the weldments are under the action of various loadings which are not only working in multiple axis direction but also time-dependent having a cyclic and mixed-mode in nature and non-relative to each other. The surface of the rail and its weldment is acted by very high repetitive stress through the wheel and because of this contact stress the running surface or subsurface may have cracks or fractures due to fatigue. This work is based on numerical simulation of an aluminum thermite weldment on a UIC 60 rail under multi-axial fatigue crack propagation under the friction with surficial interaction between weldment and wheel with bending load due to vertically applied load through the wheel on the weld. Since contact is highly influenced by vertical load and also for minimizing the simulation time the lateral and longitudinal traction forces are not included in this study. The work formulation and discretization have been done with the finite element method and a non-linear lagrangian algorithm solver is applied. A 3-D rail-weld wheel model assembly and a semi-elliptical crack as a flaw on the weld surface are used to identify 3-Modes of SIFs along with its graphical plot generation. Simulation is performed under multi-axial weld wheel surface contact at different locations on weld running surface, taking into account varying position of fracture crack on weld 3-D model to calculate fracture life of weld joint and observation of fatigue crack propagation. This work involves the numerical and theoretical approach of fracture mechanics on created FE fatigue model using the Linear Elastic Fracture Mechanics (LEFM) method following Paris law for fracture mechanics. All the numerical simulation for critical fracture dimension and cycle count with stress intensity factor for weld failure data is estimated using software ANSYS 2020 academic and plotted, then comparison of predicted and observed transverse crack growth behavior and fatigue life of weld, based on Millions Gross Tonnes (MGT) is discussed.

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Bonaventura, Luca, Elisa Calzola, Elisabetta Carlini, and Roberto Ferretti. "Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems." Journal of Scientific Computing 88, no. 1 (June 5, 2021). http://dx.doi.org/10.1007/s10915-021-01518-8.

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AbstractWe propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

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Perugia, Ilaria, Christoph Schwab, and Marco Zank. "Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDEs." ESAIM: Mathematical Modelling and Numerical Analysis, October 13, 2022. http://dx.doi.org/10.1051/m2an/2022081.

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For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete hp-time-stepping approach. We combine this semi-discretization in time with first-order, so-called "h-version" Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of D, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in D. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

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Bergami, Matteo, Walter Boscheri, and Giacomo Dimarco. "A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes." Communications on Applied Mathematics and Computation, December 11, 2020. http://dx.doi.org/10.1007/s42967-020-00093-3.

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AbstractIn this paper, we present a conservative semi-Lagrangian scheme designed for the numerical solution of 3D hydrostatic free surface flows involving sediment transport on unstructured Voronoi meshes. A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume. This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms. The presented method is fully conservative by construction, since the transported quantity or the vector field is integrated for each cell over the deformed volume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element. The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations. Furthermore, a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance, hence yielding a stability condition which depends only on the explicit discretization of the viscous terms. A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment, which is assumed to be passive. The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solution whenever is known. The new numerical scheme can reach up to fourth order of accuracy on general orthogonal meshes composed by Voronoi polygons.

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El-Amrani, Mofdi, Abdellah El-Kacimi, Bassou Khouya, and Mohammed Seaid. "Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems." Journal of Scientific Computing 92, no. 2 (July 6, 2022). http://dx.doi.org/10.1007/s10915-022-01888-7.

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AbstractA class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.

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Journal articles on the topic 'Semi-Lagrangian discretizations'

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