A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes

Busto, Saray; Río-Martín, Laura; Vázquez-Cendón, M. Elena; Dumbser, Michael

doi:10.1016/j.amc.2021.126117

Cited by 29 publications

(49 citation statements)

References 147 publications

(263 reference statements)

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“…Alternatively, other semiimplicit methods have been proposed [9,19,20,36,52] where a linearly implicit scheme is derived for the stiff terms in the governing equations, thus avoiding any need of iterative solvers. Let us mention that, semi-implicit hybrid finite volume/finite element schemes have been recently proposed in [3,14], while semi-implicit methods coupled with discontinuous Galerkin (DG) space discretizations on unstructured staggered meshes have been forwarded for compressible flows [66], on dynamic adaptive meshes [30] and for axially symmetric flows [35]. In most of the aforementioned works, the convective terms of the governing equations are discretized explicitly, because they typically involve a nonlinearity which is difficult to be implicitly solved, requiring the usage of computationally time consuming and numerically less stable nonlinear solvers for the resulting system that need to be inverted.…”

Section: Introductionmentioning

confidence: 99%

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

2022

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This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. In this way, an unconditionally stable numerical scheme is obtained, that does not suffer any CFL-type stability restriction on the maximum admissible time step. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. A conservative scheme is then crucial to properly capture the wave propagation speed and thus to locate the discontinuity and the plateau exhibited by the solution. A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy–Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.

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Section: Introductionmentioning

confidence: 99%

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

2022

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“…If we now take into account that explicit FV methods are well suited for solving hyperbolic equations and implicit FE methods are well known to be suitable and efficient for the solution of elliptic Poisson-type problems, we will end up with a hybrid FV/FE methodology that solves each subproblem with the most suitable scheme. The operator splitting between pressure-independent purely convective terms and pressure terms can also be extended to the compressible case, see [23][24][25][26][27][28], keeping a reasonable time step restriction depending only on the velocity field, rather than on the sound speed.…”

Section: Introductionmentioning

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%

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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“…Alternatively, other semi-implicit methods have been proposed [40,34,18,19,9] where a linearly implicit scheme is derived for the stiff terms in the governing equations, thus avoiding any need of iterative solvers. Let us mention that, semi-implicit hybrid finite volume/finite element schemes have been recently proposed in [13,3], while semi-implicit methods coupled with discontinuous Galerkin (DG) space discretizations on unstructured staggered meshes have been forwarded for compressible flows [48], on dynamic adaptive meshes [29] and for axially symmetric flows [33]. In most of the aforementioned works, the convective terms of the governing equations are discretized explicitly, because they typically involve a nonlinearity which is difficult to be implicitly solved, requiring the usage of computationally time consuming and numerically less stable nonlinear solvers for the resulting system that need to be inverted.…”

Section: Introductionmentioning

confidence: 99%

On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDEs

Boscheri¹,

Tavelli²,

Pareschi³

2021

Preprint

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This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. In this way, an unconditionally stable numerical scheme is obtained, that does not suffer any CFL-type stability restriction on the maximum admissible time step. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. A conservative scheme is then crucial to properly capture the wave propagation speed and thus to locate the discontinuity and the plateau exhibited by the solution. A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy-Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.

show abstract

A semi-implicit hybrid finite volume/finite element scheme for all Mach number flows on staggered unstructured meshes

Cited by 29 publications

References 147 publications

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

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

On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDEs

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