Assessment of numerical schemes for transient, finite-element ice flow models using ISSM v4.18

Time-dependent simulations of ice sheets require two equations to be solved: the mass transport equation, derived from the conservation of mass, and the stress balance equation, derived from the conservation of momentum. The mass transport equation controls the advection of ice from the interior of the ice sheet towards its periphery, thereby changing its geometry. Because it is based on an advection equation, a stabilization scheme needs to be employed when solved using the finite-element method. Several stabilization schemes exist in the finite-element method framework, but their respective accuracy and robustness have not yet been systematically assessed for glaciological applications. Here, we compare classical schemes used in the context of the finite-element method: (i) artificial diffusion, (ii) streamline upwinding, (iii) streamline upwind Petrov–Galerkin, (iv) discontinuous Galerkin, and (v) flux-corrected transport. We also look at the stress balance equation, which is responsible for computing the ice velocity that “advects” the ice downstream. To improve the velocity computation accuracy, the ice-sheet modeling community employs several sub-element parameterizations of physical processes at the grounding line, the point where the grounded ice starts to float onto the ocean. Here, we introduce a new sub-element parameterization for the driving stress, the force that drives the ice-sheet flow. We analyze the response of each stabilization scheme by running transient simulations forced by ice-shelf basal melt. The simulations are based on an idealized ice-sheet geometry for which there is no influence of bedrock topography. We also perform transient simulations of the Amundsen Sea Embayment, West Antarctica, where real bedrock and surface elevations are employed. In both idealized and real ice-sheet experiments, stabilization schemes based on artificial diffusion lead systematically to a bias towards more mass loss in comparison to the other schemes and therefore should be avoided or employed with a sufficiently high mesh resolution in the vicinity of the grounding line. We also run diagnostic simulations to assess the accuracy of the driving stress parameterization, which, in combination with an adequate parameterization for basal stress, provides improved numerical convergence in ice speed computations and more accurate results.

Elimination of Numerical Damping in the Stability Analysis of Noncompact Thermoacoustic Systems With Linearized Euler Equations

The hybrid computational fluid dynamics/computational aeroacoustics (CFD/CAA) approach represents an effective method to assess the stability of noncompact thermoacoustic systems. This paper summarizes the state-of-the-art of this method, which is currently applied for the stability prediction of a lab-scale configuration of a perfectly premixed, swirl-stabilized gas turbine combustion chamber at the Thermodynamics institute of the Technical University of Munich. Specifically, 80 operational points, for which experimentally observed stability information is readily available, are numerically investigated concerning their susceptibility to develop thermoacoustically unstable oscillations at the first transversal eigenmode of the combustor. Three contributions are considered in this work: (1) flame driving due the deformation and displacement of the flame, (2) visco-thermal losses in the acoustic boundary layer and (3) damping due to acoustically induced vortex shedding. The analysis is based on eigenfrequency computations of the Linearized Euler Equations with the stabilized finite element method (sFEM). One main advancement presented in this study is the elimination of the nonphysical impact of artificial diffusion schemes, which is necessary to produce numerically stable solutions, but falsifies the computed stability results.

A Revisit to CMFD Schemes: Fourier Analysis and Enhancement

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

Artificial diffusion in the simulation of micromixers: A review

False (artificial) diffusion provides an erroneous estimation of molecular diffusion during the simulation of liquid micromixing. The present review introduces discretization methods, numerical grid types, and numerical errors to address the effect of false diffusion on the prediction of mixing efficiency of microfluidic devices. False diffusivity is characterized by the grid Peclet number, grid type, and discretization scheme. This review demonstrates that most investigators have selected a grid resolution just for the grid independence test. It is revealed that the convergence criterion should not be quantitative values of mixing efficiency even when high-order schemes are employed to discretize the computational grid. Based on the previous publications in this field, a straightforward procedure is recommended to manage false diffusion in the numerical simulation of micromixers.

Assessment of numerical schemes for transient, finite-element ice flow models using ISSM v4.18

Time dependent simulations of ice sheets require two equations to be solved: the mass transport equation, derived from the conservation of mass, and the stress balance equation, derived from the conservation of momentum. The mass transport equation controls the advection of ice from the interior of the ice sheet towards its periphery, thereby changing its geometry. Because it is based on a hyperbolic partial differential equation, a stabilization scheme needs to be employed when solved using the finite element method. Several stabilization schemes exist in the finite element method framework, but their respective accuracy and robustness have not yet been systematically assessed for glaciological applications. Here, we compare classical schemes used in the context of the finite element method: (i) Artificial Diffusion, (ii) Streamline Upwinding, (iii) Streamline Upwind Petrov-Galerkin, (iv) Discontinuous Galerkin, and (v) Flux Corrected Transport. We also look at the stress balance equation, which is responsible for computing the ice velocity that `advects' the ice dowstream. To improve the velocity computation accuracy, the ice sheet modeling community employs several sub-element parameterizations of physical processes at the grounding line, the point where the grounded ice starts to float onto the ocean. Here, we introduce a new sub-element parameterization for the driving stress, the force that drives the ice sheet flow. We analyze the response of each stabilization scheme by running transient simulations forced by ice shelf basal melt. The simulations are based on an idealized ice sheet geometry for which there is no influence of bedrock topography. We also perform transient simulations of the Amundsen Sea Sector, West Antarctica, where real bedrock and surface elevations are employed. In both idealized and real ice sheet experiments, stabilization schemes based on artificial diffusion lead systematically to a bias towards more mass loss in comparison to the other schemes, and therefore, should be avoided or employed with a sufficiently high mesh resolution in the vicinity of the grounding line. We also run diagnostic simulations to assess the accuracy of the driving stress parameterization, which in combination with an adequate parameterization for basal stress, provides improved numerical convergence in ice speed computations and more accurate results.

Impact of the Stabilized Finite Element Method On Acoustic and Vortical Perturbations in Thermoacoustic Systems

This paper seeks to advance linear stability analyses of thermoacoustic systems conducted with the stabilized Finite Element Method (sFEM). Specifically, this work analyzes and quantifies the impact of the Streamline-Upwind-Petrov-Galerkin (SUPG) artificial diffusion scheme on (eigen)mode shapes and damping rates of the isentropic Linearized Euler Equations (LEE) in frequency space. The LEE (eigen)mode shapes are separated into acoustic and vortical perturbation components via a Helmholtz decomposition and their sensitivity on the employed stabilization scheme is investigated separately. The regions where numerical stabilization mainly acts on the perturbation types are identified and explanations for the observations are provided. A methodology is established, which allows the quantification of the impact of artificial diffusion on the acoustic field in terms of a damping rate. This non-physical damping rate is used to determine the physically meaningful, acoustic LEE damping rate, which is corrected by the contribution of artificial diffusion. Hence, the presented method eliminates a main shortcoming of LEE eigenfrequency analyses with the sFEM and, as a result, provides more accurate information on the stability of thermoacoustic systems.

Impact of the Stabilized Finite Element Method on Acoustic and Vortical Perturbations in Thermoacoustic Systems

This paper seeks to advance linear stability analyses of thermoacoustic systems conducted with the stabilized Finite Element Method (sFEM). Specifically, this work analyzes and quantifies the impact of the Streamline-Upwind-Petrov-Galerkin (SUPG) artificial diffusion scheme on (eigen)mode shapes and damping rates of the isentropic Linearized Euler Equations (LEE) in frequency space. The LEE (eigen)mode shapes are separated into acoustic and vortical perturbation components via a Helmholtz decomposition and their sensitivity on the employed stabilization scheme is investigated separately. The regions where numerical stabilization mainly acts on the perturbation types are identified and explanations for the observations are provided. A methodology is established, which allows the quantification of the impact of artificial diffusion on the acoustic field in terms of a damping rate. This non-physical damping rate is used to determine the physically meaningful, acoustic LEE damping rate, which is corrected by the contribution of artificial diffusion. Hence, the presented method eliminates a main shortcoming of LEE eigen-frequency analyses with the sFEM and, as a result, provides more accurate information on the stability of thermoacoustic systems.

Elimination of Numerical Damping in the Stability Analysis of Non-Compact Thermoacoustic Systems With Linearized Euler Equations

The hybrid Computational Fluid Dynamics/Computational AeroAcoustics (CFD/CAA) approach represents an effective method to assess the stability of non-compact thermoacoustic systems. This paper summarizes the state-of-the-art of this method, which is currently applied for the stability prediction of a lab-scale configuration of a perfectly-premixed, swirl-stabilized gas turbine combustion chamber at the Thermodynamics institute of the Technical University of Munich. Specifically, 80 operational points, for which experimentally observed stability information is readily available, are numerically investigated concerning their susceptibility to develop thermoacoustically unstable oscillations at the first transversal eigenmode of the combustor. Three contributions are considered in this work: (1) flame driving due the deformation and displacement of the flame, (2) visco-thermal losses in the acoustic boundary layer and (3) damping due to acoustically induced vortex shedding. The analysis is based on eigenfrequency computations of the Linearized Euler Equations with the stabilized Finite Element Method (sFEM). One main advancement presented in this study is the elimination of the non-physical impact of artificial diffusion schemes, which is necessary to produce numerically stable solutions, but falsifies the computed stability results.