Algebraic Flux Correction II

Kuzmin, Dmitri; Möller, Matthias; Gurris, Marcel

doi:10.1007/978-94-007-4038-9_7

Cited by 35 publications

(43 citation statements)

References 55 publications

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“…In the former equation U = (ρ, ρu, ρv, ρE) T = (U (1) , U (2) , U (3) , U (4) ) is the vector of conservative variables. The effective density is given by ρ = α p ρ p , and v = (u, v) T denotes the velocity vector.…”

Section: Mathematical Propertiesmentioning

confidence: 99%

“…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%

“…For the gaseous phase of the two-fluid model D ij are the diffusion blocks defined in [9,2,4,18]. In the case of the particulate phase D ij can be written as…”

Section: Edge-based Approximate Interior Flux Jacobianmentioning

confidence: 99%

“…To prevent the birth and growth of wiggles and preserve the physical properties of the solution, a suitable stabilization term should be added to the discretized equations. Kuzmin et al [1][2][3][4][5][6][7] have developed a new approach to the design of high-resolution finite element schemes. Within the framework of algebraic flux correction, the coefficients of a standard Galerkin discretization are constrained using flux limiters based on a generalization of flux-corrected transport (FCT) algorithms and total variation diminishing (TVD) methods.…”

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: Mathematical Propertiesmentioning

confidence: 99%

Section: Discretizationmentioning

confidence: 99%

“…For the gaseous phase of the two-fluid model D ij are the diffusion blocks defined in [9,2,4,18]. In the case of the particulate phase D ij can be written as…”

Section: Edge-based Approximate Interior Flux Jacobianmentioning

confidence: 99%

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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“…Therefore, it is hard to impose the Dirichlet boundary conditions in the usual way. It is common practice to recover the boundary values by switching to the characteristic variables, evaluating the incoming Riemann invariants from the physical boundary conditions and extrapolating the outgoing ones from the interior of the computational domain (Kuzmin & Möller, 2004).…”

Section: Boundary Conditionsmentioning

confidence: 99%

Application of Finite Volume Method in Fluid Dynamics and Inverse Design Based Optimization

Veress¹,

Rohács²

2012

Finite Volume Method - Powerful Means of Engineering Design

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Implicit finite element schemes for the stationary compressible Euler equations

Gurris

Kuzmin

Turek

2011

Numerical Methods in Fluids

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SUMMARY A semi‐implicit finite element scheme and a Newton‐like solver are developed for the stationary compressible Euler equations. Since the Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable, the troublesome antidiffusive part is constrained within the framework of algebraic flux correction. A generalization of total variation diminishing (TVD) schemes is employed to blend the original Galerkin scheme with its nonoscillatory low‐order counterpart. Unlike standard TVD limiters, the proposed limiting strategy is fully multidimensional and readily applicable to unstructured meshes. However, the nonlinearity and nondifferentiability of the limiter function makes efficient computation of stationary solutions a highly challenging task, especially in situations when the Mach number is large in some subdomains and small in other subdomains. In this paper, a semi‐implicit scheme is derived via a time‐lagged linearization of the Jacobian operator, and a Newton‐like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability, as well as higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. The overall spatial accuracy of the constrained scheme and the benefits of the new boundary treatment are illustrated by grid convergence studies for 2D benchmark problems. Copyright © 2011 John Wiley & Sons, Ltd.

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Algebraic Flux Correction II

Cited by 35 publications

References 55 publications

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

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

Application of Finite Volume Method in Fluid Dynamics and Inverse Design Based Optimization

Implicit finite element schemes for the stationary compressible Euler equations

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