Preprint on high-resolution finite element schemes for an idealized Z-pinch implosion model

31, May, 2010

The following preprint has been added to my list of publications.

D. Kuzmin, M. Möller and J.N. Shadid, High-resolution finite element schemes for an idealized Z-pinch implosion model (PS, PDF). Technical Report 410, Technische Universität Dortmund, 2010.

Abstract

A high-resolution finite element method is developed for numerical simulation of Z-pinch-like implosions using a phenomenological model for the magnetic drive source term. The momentum and energy equations of the Euler system are extended by adding radial body forces proportional to the concentration of a scalar tracer field. The evolution of the tracer is governed by an additional transport equation which is solved in a segregated fashion. The finite element discretization is stabilized using a linearized flux-corrected transport (FCT) algorithm. Scalar viscosity of Rusanov type is employed to construct the underlying low-order scheme. In the process of flux limiting, node-by-node transformations from the conservative to the primitive variables are performed to ensure that all quantities of interest (density, pressure, tracer) are bounded by the physically admissible low-order values. The performance of the proposed algorithm on fully unstructured meshes is illustrated by numerical results for a power law implosion in the x-y plane.

The paper has been submitted to Computers & Fluids.

Preprint on failsafe flux limiting

21, Apr, 2010

The following preprint has been added to my list of publications.

D. Kuzmin, M. Möller, J.N. Shadid and M. Shashkov, Failsafe flux limiting and constrained data projections for systems of conservation laws (PS, PDF). Technical Report 407, Technische Universität Dortmund, 2010.

Abstract

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L2 projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.

The paper has been submitted to Journal of Computational Physics.

Talk at ICFD Conference 2010

19, Apr, 2010

The slides of my talk On the design of high-resolution finite element schemes for coupled problems with application to an idealized Z-pinch implosion model presented at the 10th ICFD Conference on Numerical Methods for Fluid Dynamics, April 11-15, 2010, at the University of Reading, England can be downloaded here.

Talk at University of Houston, TX

23, Feb, 2010

I gave an invited talk about Failsafe Flux Limiting for Implosion Models in the Scientific Computing Seminar of the Department of Mathematics at University of Houston.

Abstract

A popular approach to numerical simulation of unsteady flow problems are flux-corrected transport (FCT) algorithms which are capable of producing accurate solutions without numerical artifacts such as non-physical oscillations in the vicinity of step gradients. In the first part of this talk, the design of algebraic flux correction (AFC) finite element schemes is revisited in the context of the compressible Euler equations of gas dynamics. A new synchronized FCT algorithm is presented which ensures that all selected quantities of interest (density, energy, pressure, velocity) are bounded by the values of their low-order counterparts. To this end, the fully multidimensional FCT flux limiter is applied to a set of indicator quantities, whereby a node-based transformation from the conservative to the primitive variables is performed. To reduce the computational costs of the FCT algorithm, the raw antidiffusive fluxes are linearized about an auxiliary state, so that the solution-dependent correction factors need to be computed just once per time step. Moreover, flux limiting reduces to a simple post-processing of the converged low-order solution. It can therefore be equipped with a simple failsafe strategy which ensures that the flux-corrected end-of-step solution is still bounded by its low-order values. In the second part of the talk, the presented techniques are applied to an idealized Z-pinch implosion model recently proposed by J.W. Banks and J.N. Shadid. It is based on the compressible Euler equations which include a magnetic source term and are coupled with a scalar tracer equation. The synchronized flux limiting procedure is shown to produce promising results for this challenging problem.

The presentation can be downloaded here.

Talk at SNL in Albuquerque, NM

16, Oct, 2009

I gave an invited talk about high-resolution finite element schemes for coupled problems at SANDIA National Laboratories.

Abstract

Algebraic criteria for the design of non-oscillatory methods are reviewed in a general framework. For a given discretization (e.g., standard Galerkin finite elements) these mathematical constraints can be readily enforced by means of conservative matrix manipulations. A family of high-resolution finite element schemes for convection-dominated flow problems is constructed based on the so-called algebraic flux correction (AFC) approach. Implicit time integration methods are employed to permit the use of moderately large time steps even on locally refined meshes.

A linearized version of the flux corrected transport (FCT) algorithm is presented which allows for an accurate cost-effective treatment of highly time-dependent flow problems. In each time step, an implicit non-oscillatory low-order method is used to predict a provisional solution. Flux limiting is applied as post-processing step to remove excessive numerical diffusion without generating spurious undershoots and overshoots. The basic concepts are first described for scalar conservation laws and extended to hyperbolic system of equations.

Numerical examples are presented for the compressible Euler equations which are equipped with a thin-shell Lorentz force model and coupled with a scalar tracer equation. All simulations are performed using the in-house code Featflow2. Some implementation details and advanced features such as dynamic grid adaptation are briefly addressed.

If you are interested in the presentation including all animations you may download this ZIP-archive. The slides without additional animation files are available here.

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