Finite element form of FDV for widely varying flowfields

  • Authors:

  • G. A. Richardson;J. T. Cassibry;T. J. Chung;S. T. Wu

  • Affiliations:

  • Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, AL 35899, United States and Center for Space Plasma and Aeronomic Research, University of Alaba ...;Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, AL 35899, United States and Propulsion Research Center, University of Alabama in Huntsville, Hu ...;Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, AL 35899, United States;Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, AL 35899, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

We present the Flowfield Dependent Variation (FDV) method for physical applications that have widely varying spatial and temporal scales. Our motivation is to develop a versatile numerical method that is accurate and stable in simulations with complex geometries and with wide variations in space and time scales. The use of a finite element formulation adds capabilities such as flexible grid geometries and exact enforcement of Neumann boundary conditions. While finite element schemes are used extensively by researchers solving computational fluid dynamics in many engineering fields, their use in space physics, astrophysical fluids and laboratory magnetohydrodynamic simulations with shocks has been predominantly overlooked. The FDV method is unique in that numerical diffusion is derived from physical parameters rather than traditional artificial viscosity methods. Numerical instabilities account for most of the difficulties when capturing shocks in these regimes. The first part of this paper concentrates on the presentation of our numerical method formulation for Newtonian and relativistic hydrodynamics. In the second part we present several standard simulation examples that test the method's limitations and verify the FDV method. We show that our finite element formulation is stable and accurate for a range of both Mach numbers and Lorentz factors in one-dimensional test problems. We also present the converging/diverging nozzle which contains both incompressible and compressible flow in the flowfield over a range of subsonic and supersonic regions. We demonstrate the stability of our method and the accuracy by comparison with the results of other methods including the finite difference Total Variation Diminishing method. We explore the use of FDV for both non-relativistic and relativistic fluids (hydrodynamics) with strong shocks in order to establish the effectiveness in future applications of this method in astrophysical and laboratory plasma environments.