Implicit finite element schemes for the stationary compressible Euler equations

Gurris, Marcel; Kuzmin, Dmitri; Turek, Stefan

doi:10.1002/fld.2532

Cited by 27 publications

(17 citation statements)

References 47 publications

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“…Hyperbolic solvers can be applied to the equations governing the gas phase, while scalar dissipation is feasible for the pressureless conservation laws of the particulate phase. The design of the artificial diffusion operator for the gas phase can be found in [4,2,9,18]. Therefore it remains to define the stabilization of the particulate phase.…”

Section: Discretizationmentioning

confidence: 99%

“…which makes it possible to prescribe boundary conditions in a weak sense [9,18]. In our algorithm, the weak form of the Galerkin discretization (27) serves as the second-order scheme, while the stabilization is still based on (24).…”

Section: Second-order Schemementioning

confidence: 99%

“…We recommend to compute the stabilization and prescribe boundary conditions separately for both phases, while the nonlinear system should be solved in a strongly coupled way. Since the treatment of the gas phase is described in [9,2,18], we focus our attention to the pressureless conservation laws of the particulate phase.…”

Section: Mathematical Propertiesmentioning

confidence: 99%

“…It may even inhibit convergence to a steady-state solution. The algorithm developed in this paper features most properties of the single-phase gas solver proposed in [9,18].…”

Section: Introductionmentioning

confidence: 99%

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Implicit finite element schemes for stationary compressible particle-laden gas flows

Gurris

Kuzmin

Turek

2011

Journal of Computational and Applied Mathematics

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a b s t r a c tThe derivation of macroscopic models for particle-laden gas flows is reviewed. Semiimplicit and Newton-like finite element methods are developed for the stationary two-fluid model governing compressible particle-laden gas flows. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers. This is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters and the additional strong nonlinearity caused by interfacial coupling terms. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. The original Jacobian is replaced by a low-order approximation. Special emphasis is laid on the numerical treatment of weakly imposed boundary conditions. It is shown that the proposed approach offers unconditional stability and faster convergence rates for increasing CFL numbers. The strongly coupled solver is compared to operator splitting techniques, which are shown to be less robust.

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Section: Discretizationmentioning

confidence: 99%

Section: Second-order Schemementioning

confidence: 99%

Section: Mathematical Propertiesmentioning

confidence: 99%

“…It may even inhibit convergence to a steady-state solution. The algorithm developed in this paper features most properties of the single-phase gas solver proposed in [9,18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Implicit finite element schemes for stationary compressible particle-laden gas flows

Gurris

Kuzmin

Turek

2011

Journal of Computational and Applied Mathematics

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“…Their construction, e.g., in [18,16,17], is performed for transport equations and they are called flux-corrected transport (FCT) schemes (see also [7] for their application to compressible flows). These schemes can be used also for the discretization of time-dependent convection-diffusion equations, e.g., as in [4,11], where the convection-diffusion equations are part of population balance systems.…”

mentioning

confidence: 99%

Analysis of Algebraic Flux Correction Schemes

Barrenechea¹,

John²,

Knobloch³

2016

SIAM J. Numer. Anal.

132

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Abstract. A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods' main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection-diffusion-reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness.Key words. algebraic flux correction method, linear boundary value problem, well-posedness, discrete maximum principle, convergence analysis, convection-diffusion-reaction equations.AMS subject classifications. 65N12, 65N301. Introduction. Many processes from nature and industry can be modelled using (systems of) partial differential equations. Usually, these equations cannot be solved analytically. Instead, only numerical approximations can be computed, e.g., by using a finite element method (FEM). The Galerkin FEM replaces just the infinite-dimensional spaces from the variational form of the differential equation with finite-dimensional counterparts. However, if the considered problem contains a wide range of important scales, the Galerkin FEM does not give useful numerical results unless all scales are resolved. For many problems, the resolution of all scales is not affordable because of the huge computational costs (memory, computing time). The remedy consists in modifying the Galerkin FEM in such a way that the effect of small scales is taken into account already on grids which do not resolve all scales. This methodology is usually called stabilization. The most common strategy modifies or enriches the Galerkin FEM, e.g., such that the new discrete problem provides additional control of the error in appropriate norms. An alternative approach acts on the algebraic level, i.e., algebraic representations of discrete operators and vectors are modified before computing a numerical solution. This paper studies a method of the latter type.Applications of algebraically stabilized FEMs can be found in particular for convection-dominated problems. Their construction, e.g., in [18,16,17], is performed for transport equations and they are called flux-corrected transport (FCT) schemes (see also [7] for their application to compressible flows). These schemes can be used also for the discretization of time-dependent convection-diffusion equations, e.g., as in [4,11], where the convection-diffusion equations are part of population balance systems. In [11] it is explicitly emphasized that the FCT scheme was preferred to

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Reduced one‐dimensional modelling and numerical simulation for mass transport in fluids

Köppl

Wohlmuth

Helmig

2012

Numerical Methods in Fluids

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SUMMARY On the basis of the Navier–Stokes equations in three space dimensions and a convection–diffusion equation, we use a nonlinear system of three hyperbolic PDEs in one space dimension to simulate mass transport. We focus on the modelling of mass transport at a bifurcation of a vessel. For the numerical treatment of the hyperbolic PDE system, we use stabilised discontinuous Galerkin (DG) approximations with a Taylor basis. DG approximations together with a suitable time integration method enable us to simulate wave propagations for many periods avoiding excessive dispersion and dissipation effects. However, standard DG approximations tend to create non‐physical oscillations at sharp fronts, and thus stabilisation techniques are required. Finally, we present some numerical results illustrating the robustness of our model and the numerical discretisation.Copyright © 2012 John Wiley & Sons, Ltd.

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Implicit finite element schemes for the stationary compressible Euler equations

Cited by 27 publications

References 47 publications

Implicit finite element schemes for stationary compressible particle-laden gas flows

Implicit finite element schemes for stationary compressible particle-laden gas flows

Analysis of Algebraic Flux Correction Schemes

Reduced one‐dimensional modelling and numerical simulation for mass transport in fluids

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