A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

Busto, Saray; Dumbser, Michael

doi:10.1016/j.apnum.2022.02.005

Cited by 25 publications

(22 citation statements)

References 143 publications

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“…To solve the incompressible RANS equations in combination with the k − ε turbulence model, we extend the family of hybrid finite volume/finite element methods described in [27][28][29][30][31][32]. This methodology relies on a specific combination of explicit and implicit FV and FE methods to solve the subsystems obtained from the flux splitting introduced in the previous section.…”

Section: The Hybrid Finite Volume/finite Element Methodsmentioning

confidence: 99%

“…To the best knowledge of the authors, this is the first time that a semi-implicit hybrid finite volume/finite element scheme is proposed on staggered Cartesian meshes, since previous work on hybrid FV/FE schemes was focused on unstructured simplex meshes in two and three space dimensions, see [27,28,[30][31][32]. Here, we propose to use a staggered Cartesian mesh similar to the one used in [8,9,12,25,88].…”

Section: Cartesian Gridsmentioning

confidence: 99%

“…A careful discretization of the governing PDE system is therefore needed, which is able to deal with both low and high Reynolds number flows. In this paper, we aim at extending the hybrid FV/FE methodology presented in the series of papers [27][28][29][30][31][32], to the case of turbulent flows and non-Newtonian power-law fluids with yield stress.…”

Section: Introductionmentioning

confidence: 99%

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Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

2021

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This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

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Section: The Hybrid Finite Volume/finite Element Methodsmentioning

confidence: 99%

Section: Cartesian Gridsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

2021

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“…In this section, we recall the mathematical models considered in this paper: the incompressible Navier-Stokes equations, the weakly compressible formulation of the Navier-Stokes equations in terms of the pressure, the fully compressible Navier-Stokes equations in terms of the total energy density, which are suitable for the description of all Mach number flows, and, last but not least, the shallow water equations for the modeling of geophysical free surface flows. Following the hybrid methodology proposed in [28][29][30][31]74], we rely on a flux vector splitting [32] of the associated semi-discrete systems in time. Consequently, we get a first set of transport-diffusion equations that can be efficiently discretized in space using classical explicit finite volume methods.…”

Section: Mathematical Models and Semi-discretization In Timementioning

confidence: 99%

“…After discretization in time and some manipulations of the resulting equations, see [74] for further details, we get…”

Section: Shallow Water Equationsmentioning

confidence: 99%

A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

2021

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In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.

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A semi‐implicit finite volume scheme for a simplified hydrostatic model for fluid‐structure interaction

Brutto

Dumbser

2022

Numerical Methods in Fluids

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Simulating fluid-structure interaction problems usually requires a considerable computational effort. In this article, a novel semi-implicit finite volume scheme is developed for the coupled solution of free surface shallow water flow and the movement of one or more floating rigid structures. The model is well-suited for geophysical flows, as it is based on the hydrostatic pressure assumption and the shallow water equations. The coupling is achieved via a nonlinear volume function in the mass conservation equation that depends on the coordinates of the floating structures. Furthermore, the nonlinear volume function allows for the simultaneous existence of wet, dry and pressurized cells in the computational domain. The resulting mildly nonlinear pressure system is solved using a nested Newton method. The accuracy of the volume computation is improved by using a subgrid, and time accuracy is increased via the application of the theta method. Additionally, mass is always conserved to machine precision. At each time step, the volume function is updated in each cell according to the position of the floating objects, whose dynamics is computed by solving a set of ordinary differential equations for their six degrees of freedom. The simulated moving objects may for example represent ships, and the forces considered here are simply gravity and the hydrostatic pressure on the hull. For a set of test cases, the model has been applied and compared with available exact solutions to verify the correctness and accuracy of the proposed algorithm. The model is able to treat fluid-structure interaction in the context of hydrostatic geophysical free surface flows in an efficient and flexible way, and the employed nested Newton method rapidly converges to a solution. The proposed algorithm may be useful for hydraulic engineering, such as for the simulation of ships moving in inland waterways and coastal regions.

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A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

Cited by 25 publications

References 143 publications

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

A semi‐implicit finite volume scheme for a simplified hydrostatic model for fluid‐structure interaction

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