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An artificial compressibility analogy approach for compressible ideal MHD

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VKI PHDT 2009-04, An artificial compressibility analogy approach for compressible ideal MHD, ISBN 978-2-930389-38-9, 156 pgs

An artificial compressibility analogy approach for compressible

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An artificial compressibility analogy approach for compressible ideal MHD
by Mehmet Sarp Yalím
PhD Thesis from the von Karman Institute/Université Libre de Bruxelles, Belgium, December 2008
ISBN 978-2-930389-38-9, 156 pgs
Abstract

Ideal magnetohydrodynamics (MHD) simulations are known to have problems in satisfying the solenoidal constraint (i.e. the divergence of magnetic field should be equal to zero,). The simulations become unstable unless specific measures have been taken.

In this thesis, a solenoidal constraint satisfying technique that allows discrete satisfaction of the solenoidal constraint up to the machine accuracy is presented and validated with a variety of test cases. Due to its inspiration from Chorin's arti cial compressibility method developed for incompressible CFD applications, the technique was named as artificial compressibility analogy (ACA) approach. It is demonstrated that ACA is a purely hyperbolic, stable and consistent
technique, which is moreover easy to implement. Unlike some other techniques, it does not pose any problems of the sort that errors accumulate in the vicinity of the stagnant regions of
flow. With these crucial properties, ACA is thought to be a remedy to the drawbacks of the most commonly used solenoidal constraint satisfying techniques in the literature namely: Incorrect shock capturing and poor performance of the convective stabilization mechanism in regions of stagnant flow for Powell's source term method; exceedingly complex implementation for constrained transport technique due to the staggered grid representation; computationally
expensive nature due to the necessity of a Poisson solver combined with hyperbolic/elliptic numerical methods for classical projection schemes.

In the first chapter of the thesis, general background knowledge is given about plasmas, MHD and its history, a certain class of upwind finite volume methods, namely Riemann solvers, and their applications in MHD, the definition, constituents, formation mechanisms and effects of space weather and some of the space missions that are and will be performed in its prediction.

Secondly, detailed analysis of the compressible ideal MHD equations is given in the form of the derivation of the equations, their dimensionless numbers which will be of use to specify the
ows in the following chapters, and finally, the presentation of the MHD waves and discontinuities, which indicates the complexity of the system of ideal MHD equations and therefore their further numerical analysis.

The next discussion is about the main subject of the thesis, namely the solenoidal constraint satisfying techniques. First of all, the definition and significance of the solenoidal constraint is given. Afterwards, the most common solenoidal constraint satisfying techniques in the literature are reviewed along with their above mentioned drawbacks. Moreover, particular emphasis is given to the Powell's source term approach which was also implemented in the upwind nite volume MHD solver developed. In addition, the hyperbolic divergence cleaning technique is presented in detail together with the resemblance and differences between it and ACA. Some other solenoidal constraint satisfying techniques are brievly mentioned at this stage. After these,
ACA is presented in the following way: The point of inspiration, which is the analogy made with Chorin's arti cial compressibility method developed for incompressible CFD, the introduction of the modified system of ideal MHD equations due to ACA, the derivation of the wave equation governing the propagation of errors and the analytical consistency proof. Having finished the core discussion of the thesis, the solver developed and its constituents are given in the fourth chapter. Furthermore, a brief overview of the platform into which this solver was implemented, namely COOLFluiD, is also given at this point.

Afterwards, a thorough numerical veri cation of the ACA approach has been made on an increasingly complex suite of test cases. The results obtained with ACA and Powell's source term implementations are given in order to numerically analyse and verify ACA and compare the two methods and validate them with the results from literature. The sixth chapter is devoted to further validation of ACA performed with a variety of more advanced space weather-related simulations. In this chapter, also the splitting technique used to treat planetary magnetosphere is presented along with its application to ACA and Powell's source term approaches. This technique is utilized in obtaining the solar wind/Earth's magnetosphere interaction results and is based on suppressing the direct inclusion of the Earth's magnetic field, which is a dipole field, in the solution variables. In this way, problems are avoided with the energy equation that could arise from the drastic change of the ratio of the dipole field and the variable eld computed by the solver (i.e.) in the computational domain. Finally, conclusions and future perspectives related to the material presented in the thesis are put forward.

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